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Oksana 18 сентября, 2017 год

Aaca eiaoeoee, l = 0, neaaoao ec ii?aaaeaiey iii?anoaa M0 = {n1 n2

· · · jklklp;l;wqqaw

?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 wq

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). q

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?q2, 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

. . . q

? 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

n1n2···nl1

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

? qq

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 eiyooeoeaioia P~t

(z) aaeeo D

m?w(~t)

P

. Anee ph?1 6= ph,

oi ?anniio?ei neaao?uaa ?acei?aiea a noiio i?inoaeoeo a?iaae

1

(nh?1 + ph?1)

uh?1 (nh?1 + ph)

uh

=

u

X

h?1

k=1

Ak

(nh?1 + ph?1)

k

+

X

uh

k=1

Bk

(nh?1 + ph)

k

,

3.1 Iauay oai?aia i ?acei?aiee e?aoiuo eioaa?aeia 46

Ak = (?1)uh?1?k

uh?1 + uh ? k ? 1

uh?1 ? k

1

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