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StoRobotov
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, . . , , . , , . , .
wewe 11.12.2017 10:10
(): 33qw3
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Q
24.11.2017 04:01
():
, . , , . - . . , . ashimo !!!! , . ashimo, , . ashimo.jp Vladimir Chulkov. - 13 500. .
!!!!!!!!!!
16.11.2017 17:17
():
, . - , , . , , :) , , . , . 5-6 , .. . , .
13.11.2017 17:11
(): 324
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,
a iauee ciaiaiaoaeu eiyo
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
13.11.2017 17:10
():
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,
a iauee ciaiaiaoaeu eiyoAaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,
a iauee ciaiaiaoaeu eiyo
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . .
13.11.2017 17:09
():
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,
a iauee ciaiaiaoaeu eiyo
fgsfgsdghs 13.11.2017 10:29
(): jyhg
Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2
nm 1}
?e(s1, s2, . . . , sm) = X
M0
1
n
s1
1
n
sl
l
.
Aiea?ai ?aaainoai (2.10) aey l m, a i?aaiiei?aiee, ?oi iii aa?ii aey
l ? 1. Ni?aaaaeeau neaao?uea au?a?aiey aey iii?anoa Ml?1 e Ml

Ml?1 = Nl ? {nl nl+1 1}, Ml = Nl ? {nl+1 nl 1}.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 30
Ion?aa iieo?aai ?aaainoai aey iii?anoa Ml?1 = NlMl e, aaeaa, ?aaai-
noai aey ?yaia
X
Ml?1
1
n
s1
1
n
sl
l
=
X
Nl
1
n
s1
1
n
sl
l
?
X
Ml
1
n
s1
1
n
sl
l
= ?(sl
, sl?1, . . . , s1) ?e(sl+1, sl+2, . . . , sm) ?
X
Ml
1
n
s1
1
n
sl
l
.
Neaaiaaoaeuii,
?e(s1, s2, . . . , sm) = X
l?1
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)l?1 X
Ml?1
1
n
s1
1
n
sl
l
=
X
l
k=1
(?1)k?1
?(sk, sk?1, . . . , s1) ?e(sk+1, sk+2, . . . , sm)
+ (?1)lX
Ml
1
n
s1
1
n
sl
l
,
?oi e o?aaiaaeinu aieacaou. I?e l = m ? 1 ?aaainoai (2.10) ?aaiineeuii
ooaa??aaie? oai?aiu, oae eae
X
Mm?1
1
n
s1
1
n
sl
l
= ?(sm, sm?1, . . . , s1).
Oai?aia aieacaia.
Ia?aeaai oaia?u e aieacaoaeunoao iaiauaiey ?aaainoaa (2.7). Iii ao-
aao ai iiiaii iioi?a ia aieacaoaeunoai Aaneeuaaa ?aaainoaa (2.6) a [2].
Iai iio?aaoaony ianeieuei aniiiiaaoaeuiuo eaii.
Ionou s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Ii?aaaeei ?enea rj = Pj
i=1 si e iiiai?eaiu
Q0 = 1,
Qk(z) = 1 ? zx1 xr1?1 + zx1 xr1 ? . . . ? zx1 xrk?1 + zx1 xrk
,
Qk = Qk(1).
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 31
Eaiia 2.7 Auiieiyaony ?aaainoai
Z
[0,1]rk
dx1dx2 dxm
Qk
= ?e(s1, s2, . . . , sk).
Aieacaoaeunoai. I?eiaiei oai?aio 2.1 e ai = 1, bi = 2
Z
[0,1]rk
dx1dx2 dxm
Qk(z)
=
Z
[0,1]rk
dx1dx2 dxm
Qk
j=1(1 ? zx1 . . . xrj
)
.
A yoii oi?aanoaa ono?aiei z e aaeieoa e ainiieucoainy eaiiie 2.5.
?anniio?ei naiaenoai eioaa?aeia
I? = 1, Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
, ? 0.
Neaanoaea 2.3 Auiieiyaony ?aaainoai Is1,s2,...,sk = Is1,s2,...,sk
(0) = ?e(s1,
s2, . . . , sk).
Aieacaoaeunoai. Yoi ia?aoi?ioee?iaea eaiiu 2.7.
Neaanoaea 2.4 Ionou ana sj 1. Oiaaa auiieiyaony ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 = ?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1).
Aieacaoaeunoai. Eiaai ?aaainoai
?
d
d? [Is1,s2,...,sk
(?)]
?=0 =
Z
[0,1]rk
?
ln(1 ? Qk)
Qk
dx1 dxrk
=
Z
[0,1]rk+1
dx0dx1 dxrk
1 ? x0Qk
Aicii?iinou aeooa?aioe?iaaiey ii ia?aiao?o ? aaao ?aaiiia?iay noi-
aeiinou eioaa?aea
Z
[0,1]rk
ln(1 ? Qk)(1 ? Qk)
?
Qk
dx1 dxrk
.
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 32
i?e ? 0. Oaia?u naaeaai a eioaa?aea caiaio xrk > 1 ? xrk e i?aanoa-
aei 1 ? x0Qk(x1, x2, . . . , 1 ? xrk
) a aeaa (aiaaaeyy e au?eoay iaeioi?ua
neaaaaiua)
1 ? x0 + x0x1 ? x0x1 + x0x1x2 ? ? x0x1 xr1?2 + x0x1 xr1?1
? x0x1 xr1 + x0x1 xr1+1 ? ? x0x1 xr2?2 + x0x1 xr2?1
. . .
? x0x1 xrk?1 + x0x1 xrk?1+1 ? ? x0x1 xrk?2 + x0x1 xrk?1
? x0x1 xrk?1 + x0x1 xrk
e i?eiaiei eaiio 2.7. Neaanoaea aieacaii.
Aaaaai
??(s1, s2, . . . , sl) = X
n1n2nl1
1
(n1 + ?)
s1 (nl + ?)
sl
,
aaa s1 1, s2, . . . , sk  iaoo?aeuiua ?enea. Yoio ?ya ?aaiiia?ii noiaeony
i?e ? 0.
Eaiia 2.8 I?e sj 1 auiieiyaony ?aaainoai
Is1,s2,...,sk
(?) = X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)?e(sj+1, sj+2, . . . , sk). (2.11)
Aieacaoaeunoai. Eiaai oi?aanoai
Qk(x1, x2, . . . , xks) = 1 ? x1 xs1?1(1 ? xs1Qk?1(xs1+1, xs1+2, . . . , xrk
)).
Aey e?aoeinoe iaicia?ei Q0 = Qk?1(xs1+1, xs1+2, . . . , xrk
). ?acei?ei a
iiauioaa?aeuiii au?a?aiee 1Qk ii noaiaiyi 1 ? Qk (aioo?e eoaa ei-
oaa?e?iaaiey 0 Qk 1)
Is1,s2,...,sk
(?) = Z
[0,1]rk
(1 ? Qk)
?
Qk
dx1 dxrk
=
Z
[0,1]rk
X
?
n=0
(1 ? Qk)
n+?
dx1 dxrk
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 33
=
X
?
n=0
Z
[0,1]rk
(x1 xs1?1(1 ? xs1Q
0
))n+?
dx1 dxrk
.
Oae eae (1?Qk)
n+? iaio?eoaoaeuii, oi aicii?iinou ia?anoaiiaee eioaa?a-
ea e noiiu aa?aioe?oaony oai?aiie Ooaeie (ni, iai?eia?, [14, aeaaa V,
6, Oai?aia 5 e caia?aiea e iae]). Oai?aia Ooaeie aiai?eo i ia?anoaiiaea
aaoo eioaa?aeia (Eaaaaa), iaiaei aaneiia?io? noiio ii?ii i?aanoaaeou
a aeaa ianianoaaiiiai eioaa?aea
X
?
n=0
an =
Z ?
0
f(t) dt,
aaa f(t) = an i?e t ? [n, n + 1). I?ieioaa?e?oai ii ia?aiaiiui x1, x2,
. . . , xs1
.
Is1,s2,...,sk
(?) = X
?
n=1
1
(n + ?)
s1
Z
[0,1]rk?s1
1 ? (1 ? Q0
)
n+?
Q0
dxs1+1 dxrk
= ??(s1)Is2,s3,...,sk ?
X
?
n=1
1
(n + ?)
s1
Is2,s3,...,sk
(n + ?). (2.12)
Aoaai aieacuaaou ooaa??aaiea eaiiu ii eiaoeoee. I?iaa?ei aaco aey
k = 1
Is1
(?) = X
?
n=1
1
(n + ?)
s1
= ??(s1).
I?aaiiei?ei, ?oi ooaa??aaiea eaiiu aieacaii aey k ? 1, aiea?ai aai
aey k. Iianoaaeyy a (2.12) aianoi Is2,s3,...,sk
(n + ?) au?a?aiea, aa?iia ii
i?aaiiei?aie? eiaoeoee, iieo?aai
Is1,s2,...,sk
(?) = ??(s1)Is2,s3,...,sk
?
X
?
n=1
1
(n + ?)
s1
X
k?1
j=1
(?1)j?1
?n+?(sj+1, sj
, . . . , s2)Isj+2,sj+3,...,sk
= ??(s1)Is2,s3,...,sk ?
X
k?1
j=1
(?1)j?1
??(sj+1, sj
, . . . , s1)Isj+2,sj+3,...,sk
=
X
k
j=1
(?1)j?1
??(sj
, sj?1, . . . , s1)Isj+1,sj+2,...,sk
,
2.4 I?iecaiayuea ooieoee aey cia?aiee acaoa-ooieoee 34
?oi, o?eouaay neaanoaea 2.3, e aieacuaaao eaiio.
Oai?aia 2.7 I?e sj 1 aa?ii ?aaainoai
?e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1)
=
X
k
j=1
(?1)j?1X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1)?e(sj+1, sj+2, . . . , sk).
Aieacaoaeunoai. I?iaeooa?aioe?oai ii ? ?aaainoai (2.11) e iianoaaei
? = 0
d
d? [Is1,s2,...,sk
(?)]
?=0
=
X
k
j=1
(?1)j?1
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 ?e(sj+1, sj+2, . . . , sk).
Ii neaanoae? 2.4 eaaay ?anou ?aaia
??e(2, {1}s1?2, 2, {1}s2?2, 2, {1}sk?2, 1),
a ec ii?aaaeaiey ??(sj
, sj?1, . . . , s1) e aa ?aaiiia?iie noiaeiinoe i?e ? 0
neaaoao, ?oi
d
d? [??(sj
, sj?1, . . . , s1)]
?=0 = ?
X
j
l=1
sl?(sj
, sj?1, . . . , sl + 1, . . . , s1).
Ioeoaa iieo?aai ooaa??aaiea oai?aiu.
Ec oai?aiu 2.7 i?e k = 1 neaaoao, ?oi ?e(2, {1}s?1) = s?(s + 1), a i?e
k = 2,
?e(2, {1}s1?2, 2, {1}s2?1) = s1?(s1 + 1)?(s2) ? s2?(s2 + 1, s1) ? s1?(s2, s1 + 1)
= s1?(s1 + s2 + 1) + s1?(s1 + 1, s2) ? s2?(s2 + 1, s1).
A neo?aa ?aaiuo sj (ionou sj = s aey e?aiai j) oaaaony iin?eoaou
i?aao? ?anou aey e?auo k.
Oai?aia 2.8 I?e iaoo?aeuiuo k, s 2 auiieiyaony ?aaainoai
?e({2, {1}s?2}k, 1) = s?(sk + 1).
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 35
Aieacaoaeunoai. ?anniio?ei i?iecaiayuea ooieoee
f?(x) = X
?
k=0
(?1)k
??({s}k)x
k =
Y
?
j=1

1 ?
x
(j + ?)
s

e g(x) = P?
k=0 ?e({s}k)x
k
. Ec eaiiu 2.8 neaaoao, ?oi
f?(x)g(x) = 1 +X
?
k=1
(I{s}k ? I{s}k
(?))x
k
. (2.13)
I?e ? = 0 iieo?aai f0(x)g(x) = 1, ioeoaa
g(x) = 1f0(x) = Y
?
j=1

1 ?
x
j
s
?1
e iu, n iiiiuu? neaanoaey 2.3, iieo?aai oai?aio 2.5.
I?iaeooa?aioe?oai oi?aanoai (2.13) ii ? e iianoaaei ? = 0. Iieu-
coynu neaanoaeai 2.4, iaoiaei
X
?
k=1
?e({2, {1}s?2}k, 1)x
k = g(x)
d
d? [f?(x)]?=0
=
Y
?
j=1

1 ?
x
j
s
?1
d
d? Y
?
j=1

1 ?
x
(j + ?)
s
#
?=0
=
X
?
j=1
1
1 ?
x
j
s
sx
j
s+1 =
X
?
k=1
s?(sk + 1)x
k
.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-
cia?aiee
E?aoiua acaoa-cia?aiey aeoeaii eco?a?ony, iaiaei aieuoeinoai ?a-
coeuoaoia i?aanoaaey?o niaie ?acee?iua oi?aanoaa ia?ao yoeie cia?a-
ieyie. A yoii ?acaaea iu einiainy eo a?eoiaoe?aneeo naienoa.
N?aae anao aaeoi?ia n iaoo?aeuiuie eiiiiiaioaie auaaeei neaao?-
uea iii?anoaa
B = {~s si ? {2, 3}}, Bw = {~s ? B w(~s) = w}.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 36
Oiooiai ([33]) auaaeioe neaao?uea aeiioacu.
Aeiioaca 1. I?e e?aii ~s0 cia?aiea ?(~s0) i?aanoaaeyaony a aeaa
eeiaeiie oi?iu n ?aoeiiaeuiuie eiyooeoeaioaie io cia?aiee ?(~s),
~s ? Bw( ~s0)
.
Yoa aeiioaca auea i?iaa?aia aey ~s0 n aanii 6 16.
Aeiioaca 2. Ana cia?aiey ?(~s), ~s ? B e 1 eeiaeii iacaaeneiu iaa Q.
Anee aeiioaca 2 aa?ia, oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu ec
aeiioacu 1 aaeinoaaiii. Ec yoeo aaoo aeiioac neaaoao, ?oi ?acia?iinou
eeiaeiiai i?ino?ainoaa, ii?i?aaiiiai e?aoiuie acaoa-cia?aieyie aana
w ?aaia dw, aaa ?enea dw ii?aaaey?ony i?iecaiayuae ooieoeae
X
?
w=0
dwx
w =
1
1 ? w2 ? w3
.
Oae eae ?({2}k) = ?
2k(2k + 1)!, oi yoe cia?aiey e??aoeiiaeuiu (e
aa?a eeiaeii iacaaeneiu iaa Q ia?ao niaie e 1). Oae?a, ii oai?aia
Aia?e, e??aoeiiaeuii ?enei ?(3). Ioiineoaeuii a?eoiaoe?aneeo naienoa
?(~s) i?e a?oaeo ~s ? B ieeaeie ii?aaaeaiinoe iiea iao.
Ionou eaeia-oi ?(~s0) ? Q, w(~s0)  ia?aoii. Anee ?(~s0)?(2k) i?aanoaa-
eyaony a aeaa eeiaeiie eiiaeiaoee n ?aoeiiaeuiuie eiyooeoeaioaie
?enae ?(~s), ~s ? Bw( ~s0)+2k (a oae e aie?ii auou ii aeiioaca 1), oi neaai-
aaoaeuii n?aae yoeo ?enae anou oioy au iaii e??aoeiiaeuiia. Iai?eia?,
anee ?(2, 3) ? Q eee ?(3, 2) ? Q, oi iaii ec ?enae ?(3, 2, 2), ?(2, 3, 2) e
?(2, 2, 3) e??aoeiiaeuii. Aiaeiae?ii, anee eaeia-oi ?(~s0) ? Q, w(~s0) 
?aoiia e ?(~s0)?(3) i?aanoaaeyaony a aeaa eeiaeiie eiiaeiaoee n ?aoei-
iaeuiuie eiyooeoeaioaie ?enae ?(~s), ~s ? Bw( ~s0)+3, oi n?aae ieo anou oioy
au iaii e??aoeiiaeuiia.
Aaeaa iu aiea?ai iaeioi?ue ?acoeuoao i eeiaeiie iacaaeneiinoe
e?aoiuo acaoa-cia?aiee.
Eaiia 2.9 Ionou x ? Q, ?enea yi
, i = 1, . . . , k  oaeea, ?oi 1, y1, .. . ,
yk eeiaeii iacaaeneiu iaa Q. Oiaaa nouanoao?o k ?1 ?enae ec xyi
, ?oi
1, x e iie eeiaeii iacaaeneiu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 37
Aieacaoaeunoai. Aoaai aieacuaaou io i?ioeaiiai. Ionou ?enea 1, x, xyi
,
i = 1, . . . , k ?1 eeiaeii caaeneiu iaa Q. O.a. nouanoao?o oaeea oaeua A1,
B1 e C1i
, ia ?aaiua iaiia?aiaiii ioe?, ?oi
A1 + B1x +
X
k?1
i=1
C1ixyi = 0.
Anee A1 = 0, oi iiaaeea yoi ?aaainoai ia x, iieo?ei, ?oi 1 e ?enea yi
, i =
1, . . . , k ?1 eeiaeii caaeneiu, ?oi ii oneiae? ia oae. Anee au ana C1i = 0,
oi x auei au ?aoeiiaeuiui. Neaaiaaoaeuii, nouanoaoao p ? [1, k ?1], ?oi
C1p 6= 0. Ionou oaeua A2, B2 e C2i
, ia ?aaiua iaiia?aiaiii ioe? oaeiau,
?oi
A2 + B2x +
X
16i6k,i6=p
C2ixyi = 0.
Aiaeiae?ii, A2 6= 0. Oiii?ei ia?aia ?aaainoai ia A2 e au?oai aoi?ia
?aaainoai, oiii?aiiia ia A1. Iieo?ei (iieaaay C1k = 0, C2p = 0)
(B1A2 ? B2A1)x +
X
k
i=1
(C1iA2 ? C2iA1)xyi = 0.
Iiaaeei yoi ?aaainoai ia x. Oiaaa iieo?ei eeiaeio? oi?io io 1, yi
, i?e-
?ai eiyooeoeaio i?e yp aoaao ?aaai C1pA2 6= 0, i?ioeai?a?ea n eeiaeiie
iacaaeneiinou? 1 e ?enae yi
. Eaiia aieacaia.
Neaanoaea 2.5 I?e e?aii iaoo?aeuiii l ?enea 1, ?(3) e eaeea-oi l
?enae ec ?(3)?(2k), k = 1, . . . , l + 1 eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. A eaiia 2.9 aicuiai x = ?(3), yk = ?(2k).
Ec yoiai neaanoaey auoaeaao a?oaia
Neaanoaea 2.6 Anee Mw - iii?anoai aaeoi?ia aana w oaeeo, ?oi ana
e?aoiua acaoa-ooieoee aana w au?a?a?ony ?aoeiiaeuiui ia?acii ?a-
?ac ?(~s), ~s ? Mw, oi nouanoao?o l oaeeo aaeoi?ia ~ti ?aciiai aana,
i ? {5, 7, . . . , 2l + 5}, ~ti ? Mi
, ?oi 1, ?(3) e ?enea ?(~ti) eeiaeii iacaaene-
iu iaa Q.
2.5 A?eoiaoe?aneea naienoaa e?aoiuo acaoa-cia?aiee 38
Ii aeiioaca 1 a ea?anoaa Mw ii?ii acyou Bw. Anee oae, oi
dimQ(Q ?
M
~s?B3??B2l+5
Q?(~s)) l + 2.
Oae?a, i?aaeaii,
dimQ(Q ?
M
~s?B2??B2l
Q?(~s)) l + 1.
Neaanoaea 2.7 Nouanoaoao oaeia
~s0 ? {(2, 3),(3, 2),(2, 2, 3),(2, 3, 2),(3, 2, 2)},
?oi ?enea 1, ?(3) e ?(~s0) eeiaeii iacaaeneiu iaa Q.
Aieacaoaeunoai. I?eiaiei neaanoaea 2.6 i?e l = 1, auae?ay M5 =
{(2, 3),(3, 2)} e M7 = {(2, 2, 3),(2, 3, 2),(3, 2, 2)}.
Aeaaa 3 ?acei?aiey e?aoiuo eioaa?aeia a eeiaeiua oi?iu 39
Aeaaa 3
?acei?aiey e?aoiuo
eioaa?aeia a eeiaeiua
oi?iu
O?a eeanne?aneei ?acoeuoaoii yaeyaony i?aanoaaeaiea aeia?aaiiao-
?e?aneiai eioaa?aea
Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
(1 ? zx1x2 . . . xm)
a0
dx1dx2 . . . dxm
i?e iaoo?aeuiuo ai
, bi a aeaa Pm
s=0 Ps(z
?1
) Lis(z) (ni., iai?eia?, [16, Proposition
1, Lemma 1, Lemma 2]). Caanu e aaeaa eiyooeoeaiou i?e (iaia-
uaiiuo) iieeeiaa?eoiao a ?acei?aiee eioaa?aeia  iiiai?eaiu n ?a-
oeiiaeuiuie eiyooeoeaioaie.
A ?aaioao [20], [21] A.I. Ni?ieei ii nouanoao aieacae oi?aanoaa
Z
[0,1]3
x
n
1
(1 ? x1)
nx
n
2
(1 ? x2)
nx
n
3
(1 ? x3)
n
(1 ? zx1x2)
n+1(1 ? zx1x2x3)
n+1 dx1dx2dx3 (3.1)
= P2,1(z
?1
) Le2,1(z) + P1,1(z
?1
) Le1,1(z) + P1(z
?1
) Le1(z) + P?(z
?1
)
e
Z
[0,1]2l
Q2l
i=1 x
ai?1
i
(1 ? xi)
n
Ql
j=1(1 ? zx1x2 . . . x2j )
n+1
dx1dx2 . . . dx2l (3.2)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 40
=
X
l
k=0
Pk(z
?1
) Li{2}k
(z) +X
l?1
k=0
Tk(z
?1
) Li1,{2}k
(z),
aaa a2j?1 = a2j = (l + 1 ? j)(n + 1) ? ?, 0 6 ? 6 l 6 n. Nouanoaiaaiea
oaeiai ?acei?aiey auei iieacaii n iiiiuu? aii?ieneiaoee Iaaa.
A aaiiie aeaaa iu eco?ei iaiauaiea yoeo oaeoia, a eiaiii ?acei?a-
iea eioaa?aea
S(z) = Z
[0,1]m
Qm
i=1 x
ai?1
i
(1 ? xi)
bi?ai?1
Ql
j=1(1 ? zx1x2 . . . xrj
)
cj
dx1dx2 . . . dxm,
0 = r0 r1 r2 rl = m.
a eeiaeiua oi?iu io iaiauaiiuo iieeeiaa?eoiia. Aoaoo eniieuciaaou-
ny neaao?uea iaicia?aiey. Aoaai ienaou, ?oi ~u 6 ~v, anee aeeiu yoeo
aaeoi?ia ?aaiu e ui 6 vi i?e e?aii i = 1, . . . , l(~u) = l(~v). Iaciaai aaeoi?
~u iia?eiaiiui aaeoi?o ~v, anee ~u 6 ~v eee ~u 6 v~0 aey iaeioi?iai aaeoi?a
v~0
, iieo?aiiiai ec aaeoi?a ~v au?a?eeaaieai ianeieueeo eiiiiiaio a i?i-
ecaieuiuo ianoao. Aunioie iiiai?eaia iaciaai iaeneioi iiaoeae aai
eiyooeoeaioia.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo
eioaa?aeia
Eaiia 3.1 Iaiauaiiua iieeeiaa?eoiu Les1,s2,...,sn
(z) n ?acee?iuie ia-
ai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z).
Aieacaoaeunoai. Ecaanoii, ?oi iaiauaiiua iieeeiaa?eoiu Lis1,s2,...,sn
(z)
n ?acee?iuie iaai?aie eiaaenia eeiaeii iacaaeneiu iaa C(z) (ni. [37],
[23]). Iaai?u ooieoee {Le~s(z)} e {Li~s(z)} n w(~s), ia i?aainoiayuei iaei-
oi?iai oeene?iaaiiiai ?enea e oii?yai?aiiuo ii aic?anoaie? aeeiu
~s, naycaiu i?aia?aciaaieai c aa?oiao?aoaieuiie iao?eoae n iaioeaauie
aeaaiiaeuiuie yeaiaioaie (ni. [23, ioieo 3])
Le~s(z) = Li~s(z) +X
~t
Li~t
(z),
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 41
aaa aaeoi?a ~t a noiia eia?o oio ?a aan, ?oi e ~s, ii iaiuoo? aeeio. Ioeoaa
e neaaoao eeiaeiay iacaaeneiinou Le~s(z) iaa C(z).
Neaanoaea 3.1 Anee ooieoey f(z) eiaao i?aanoaaeaiea a aeaa eiia?-
iie noiiu P
~s P~s(z
?1
) Le~s(z), P~s(x)  iiiai?eaiu, oi yoi i?aanoaaeaiea
aaeinoaaiii.
Ii?aaaeei eiaaen ?aoeiiaeuiie ooieoee R(x) = P(x)
Q(x)
eae I(R) =
deg P ? deg Q. Ooieoee R(?1, ?2, . . . , ?l) = R1(?1) Rl(?l) io ianeieueeo
ia?aiaiiuo niiinoaaei aaeoi? ec eiaaenia (I(R1), . . . , I(Rl)).
Oai?aia 3.1 Ionou aey ooieoee R(?1, ?2, . . . , ?l) = R1(?1). . . Rl(?l) au-
iieiyaony ia?aaainoai I(R1) + I(R2) + + I(Rj ) + j 6 0 aey e?aiai
j = 1, . . . , l e ana iie?na Rj ea?ao a iii?anoaa {0, ?1, ?2, . . . }. I?e
yoii iaicia?ei mj  iaeneiaeuiue ec ii?yaeia yoeo iie?nia, p e P
 niioaaonoaaiii ieieiaeuiia e iaeneiaeuiia cia?aiey aanie?oiuo
aaee?ei iie?nia anao ooieoee Rj
.
Oiaaa i?e z ? C, z 1 noiia
X
n1n2...nl1
R(n1, n2, . . . , nl)z
n1?1
(3.3)
i?aanoaaeyaony a aeaa
X
~s
P~s(z
?1
) Le~s(z), (3.4)
aaa noiie?iaaiea aaaaony ii aaeoi?ai ~s, oaiaeaoai?y?uei oneiae?
~s 6 (m1 ? m2 ? ? ml), aaa '' icia?aao eeai caiyoo?, eeai ie?n i?e
eaeii-eeai eo ?ani?aaaeaiee (a ?anoiinoe, aoaoo auiieiyouny ia?a-
aainoaa l(~s) 6 l e w(~s) 6 m1 + m2 + + ml), a P~s(x)  iiiai?eaiu n
?aoeiiaeuiuie eiyooeoeaioaie oaeea, ?oi
ord
z=0
P?(z) 1, ord
z=0
P~s(z) p + 1 i?e ~s 6= ?, deg P~s(x) 6 P + 1.
Aiiieieoaeuii, anee auiieiy?ony ia?aaainoaa
I(R1) + I(R2) + + I(Rj ) + j 6 ?1, j = 1, . . . , l, (3.5)
oi P~s(1) = 0, aey aaeoi?ia ~s n s1 = 1.
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 42
Aiea?ai aia?aea neaao?uo? eaiio.
Eaiia 3.2 Ionou l  iaoo?aeuiia ?enei e oai?aia 3.1 aa?ia aey ooie-
oee R(?1, ?2, . . . , ?r) = R1(?1) Rr(?r) i?e r l (a neo?aa l = 1 ieeaeeo
i?aaiiei?aiee ia o?aaoaony). Oiaaa oai?aia aa?ia aey R(?1, ?2, . . . , ?l) =
R1(?1)R2(?2). . . Rl(?l), Rj (x) = 1
(x+pj )
uj
. Oneiaea (3.5) a yoii neo?aa ?aa-
iineeuii u1 2. Auniou iiiai?eaiia P~s ia i?aainoiayo
max(l! (w(~u)2w(~u)
)
l?1P
l
, 1) (3.6)
e D
w(~u)?w(~s)
P P~s(z) ? Z[z].
Aieacaoaeunoai. O?aaoaony aieacaou oai?aio 3.1 aey noiiu
X
n1n2...nl1
z
n1?1Y
l
j=1
1
(nj + pj )
uj
, (3.7)
i?e?ai min
16j6l
pj = p, max
16j6l
pj = P. Oaeea noiiu aoaai aaeaa iacuaaou
yeaiaioa?iuie. Ionou r0 = 0, rj = u1 + u2 + + uj
, m = rl = w(~u).
Eniieucoy eaiio 2.1, au?a?aiea (3.7) ii?ii caienaou a aeaa eioaa?aea
I(p1, p2, . . . , pl) = Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
I?iaaaai eiaoeoe? ii aaee?eia p1 + p2 + + pj
. I?e yoii iiea?ai
oieuei, ?oi noiia (3.7) i?aanoaaeia a aeaa (3.4), oae eae a ea?aii ec
?acae?aaiuo neo?aaa iao?oaii i?ineaaeou ca noaiaiyie iiiai?eaiia, a
oae?a ca ia?aie?aieai ia aaeoi?a iieo?a?ueony iaiauaiiuo iieeeiaa-
?eoiia.
Aaca eiaoeoee (p1 = p2 = = pl = 0) neaaoao ec eaiiu 2.2 I(0, 0, . . . ,
0) = z
?1 Leu1,u2,...,ul
(z).
?anniio?ei neo?ae pj 0 aey e?aiai j = 1, . . . , l. Ec ?aaainoaa
x1x2 . . . xrl =
1 ? (1 ? zx1x2 . . . xrl
)
z
neaaoao, ?oi
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 43
?z
?1
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj?1
Ql?1
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
A iineaaiai eioaa?aea i?ieioaa?e?oai ii ia?aiaiiui xrl?1+1, xrl?1+2, . . . ,
xrl e iieo?aiiue eioaa?ae ?acei?ei a noiio ii eaiia 2.1
I(p1, p2, . . . , pl) = z
?1
I(p1 ? 1, p2 ? 1, . . . , pl ? 1)
? z
?1

1
p
ul
l

X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj ? 1)uj
.
Eioaa?ae I(p1 ? 1, p2 ? 1, . . . , pl ? 1) i?aanoaaeyaony a aeaa (3.4) ii i?aa-
iiei?aie? eiaoeoee, a au?eoaaiay noiia i?aanoaaeyaony a aeaa (3.4) ii
oneiae? eaiiu (iia caaeneo io l ? 1 ia?aiaiiie). Oaeei ia?acii ii?ii
n?eoaou p = min
16j6l
pj = 0.
Ionou oaia?u ph 0 i?e iaeioi?ii h 1. Caieoai ?aaainoai
(xrh?1+1xrh?1+2 . . . xrh
)
ph = (xrh?1+1xrh?1+2 . . . xrh
)
ph?1
+(xrh?1+1xrh?1+2 . . . xrh
)
ph
(1 ? zx1x2 . . . xrh?1
)
?(xrh?1+1xrh?1+2 . . . xrh
)
ph?1
(1 ? zx1x2 . . . xrh
),
ec eioi?iai neaaoao
I(p1, p2, . . . , ph, . . . , pl) = I(p1, p2, . . . , ph ? 1, . . . , pl)
+
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
pj
Ql
j=1
j6=h?1
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
?
Z
[0,1]m
Ql
j=1(xrj?1+1xrj?1+2 . . . xrj
)
p
0
j
Ql
j=1
j6=h
(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm,
aaa p
0
j = pj i?e j 6= h e p
0
h = ph ? 1. Eniieucoy eaiio 2.1, ia?aieoai yoi
?aaainoai eae
I(p1, p2, . . . , ph, . . . , pl)
= I(p1, p2, . . . , ph ? 1, . . . , pl) (3.8)
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 44
+
X
n1n2...nl?11
z
n1?1
h
Y?2
j=1
1
(nj + pj )
uj
?
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh

Y
l?1
j=h
1
(nj + pj+1)
uj+1
(3.9)
?
X
n1n2...nl?11
z
n1?1
h
Y?1
j=1
1
(nj + pj )
uj
?
1
(nh + ph ? 1)uh(nh + ph+1)
uh+1

Y
l?1
j=h+1
1
(nj + pj+1)
uj+1
(3.10)
A neo?aa h = l au?eoaaiay noiia auaeyaeo eae
1
p
ul
l
X
n1n2...nl?11
z
n1?1Y
l?1
j=1
1
(nj + pj )
uj
E I(p1, p2, . . . , ph ? 1, . . . , pl) i?eiaieii i?aaiiei?aiea eiaoeoee, a aaa
a?oaea noiiu ii oneiae? eaiiu i?aanoaaey?ony a aeaa (3.4).
Inoaaony aieacaou ooaa??aaiea eaiiu aey eioaa?aea
I(p1, 0, . . . , 0) = Z
[0,1]m
(x1x2 . . . xr1
)
p1
Ql
j=1(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm.
Ec ?aaainoaa
(x1x2 . . . xr1
)
p1 = z
?1
(x1x2 . . . xr1
)
p1?1 ?z
?1
(x1x2 . . . xr1
)
p1?1
(1?zx1x2 . . . xr1
)
neaaoao
I(p1, 0, . . . , 0) = z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1
Z
[0,1]m
(x1x2 . . . xr1
)
p1?1
Ql
j=2(1 ? zx1x2 . . . xrj
)
dx1dx2 . . . dxm
= z
?1
I(p1 ? 1, 0, . . . , 0)
? z
?1 X
n1...nl?11
z
n1?1
1
(n1 + p1 ? 1)u1n
u2
1
Y
l?1
j=2
1
n
uj+1
j
,
3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 45
Au?eoaaiay noiia ii oneiae? eaiiu, a I(p1 ? 1, 0, . . . , 0) ii i?aaiiei-
?aie? eiaoeoee, i?aanoaaey?ony a aeaa (3.4). I?aanoaaeaiea a aeaa (3.4)
oaia?u iieiinou? aieacaii.
Ia?aeaai oaia?u e ioaiea aunio e a?eoiaoe?aneei naienoaai eiyo-
oeoeaioia iiiai?eaiia P~s(z). Ooaa??aaiea, eioi?ia iu aoaai aieacu-
aaou ii eiaoeoee, iaiiiai aieaa no?iaia, ?ai ooaa??aaiea eaiiu auniou
P~s(z) ia i?aainoiayo
max X
l
j=1
pj
(l ? 1)! (m2
mP)
l?1
, 1
!
.
Yoi ioaiea aaenoaeoaeuii aieaa oi?iay, ?ai (3.6), oae eae Pl
j=1 pj 6 l P.
Aieacaoaeunoai i?iaaaai eiaoeoeae ii aaeoi?o (l, p1 + p2 + + pl).
Aaeoi?a (l, k) iu oii?yai?eaaai a eaeneeia?aoe?aneii ii?yaea, o.a.
(l1, k1) (l2, k2) ? l1 l2 eee l1 = l2 e k1 k2.
Aaca eiaoeoee ni?aaaaeeaa anee pj = 0 aey anao j, oi enoiaiay noi-
ia ?aaia z
?1 Leu1,u2,...,ul
(z). Ionou oaia?u nouanoaoao pj 0 (a cia?eo e
P 0). Oiaaa i?iaaeaai oa ?a naiua i?aia?aciaaiey, ?oi auee auoa
(iaiiiiei, ?oi i?aanoaaeaiea a aeaa eeiaeiie oi?iu (3.4) aaeinoaaiii
ii neaanoae? 3.1). A ea?aii ec o?ao neo?aaa aieacaoaeunoaa aiaeiae?iu,
iiyoiio ?acaa?ai oieuei aoi?ie neo?ae (eiaaa ph 0 i?e h 1).
?anniio?ei iia?iaiaa noiio (3.9). Anee ph?1 = ph, oi
1
(nh?1 + ph?1)
uh?1 (nh?1 + ph)
uh
=
1
(nh?1 + ph?1)
uh?1+uh
,
o.a. noiia (3.9) naia yaeyaony yeaiaioa?iie e e iae ii?ii i?eiaieou
i?aaiiei?aiee eiaoeoee. A yoii neo?aa auniou iiiai?eaiia P~t
(z) a a?
?acei?aiee ia i?aainoiayo
(l ? 1)! (m2
m)
l?2P
l?1
,
a iauee ciaiaiaoaeu eiyo
1001 10.11.2017 01:27
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