Method for obtaining combinatorial identities withpolynomial coefficientsby means of integral representations Текст научной статьи по специальности «Математика»

УДК 517.55+519.1

Method for Obtaining Combinatorial Identities with Polynomial Coefficients by Means of Integral Representations

Viacheslav P. Krivokolesko*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 06.12.2015, received in revised form 08.01.2016, accepted 12.02.2016 We propose a method how to derive combinatorial identities with polynomial coefficients by means of an integral representation of holomorphic functions in an n-circular linearly convex polyhedron.

Keywords: combinatorial identities, polynomial coefficients, integral representation. DOI: 10.17516/1997-1397-2016-9-2-192-201.

Introduction

In [1] a number of some combinatorial identities was obtained. One of such identities is a generalization of the Chaundy-Bullard identity [2], see also [3-10]

The proof was based on integration of holomorphic monomials along a piecewise regular boundary of a bounded linearly convex n-circular domain in Cn, in particular, of a bicircular domain in C2.

Similar identities were obtained in [11], where holomorphic monomials were integrated along the boundary of a 3-circular domain in C3.

These identities were verified and generalized by the method of Egorychev from [12] developed in [13].

The domains considered in [1] and [11] were not complete and the identities obtained have binomial and polynomial coefficients.

The structure of the integral representation [14] and known examples imply the following conjecture

Problem 1. The integral representation in [14] permits one to obtain combinatorial identities related to the geometry of a domain, along whose boundary the integration of a holomorphic function is performed.

As follows from the proof of theorem 6 of this paper, this conjecture is proved to be true for complete bounded linearly convex domains with piecewise regular boundary in Cn.

The article consists of three sections, the first two preceed the proof of the main theorem.

In the first section we formulate the theorem on the integral representation in bounded n-circular linearly convex domains with piecewise regular boundary in Cn and introduce the necessary notation and definitions.

In the second section we study some properties of the projection n : Cn ^ R+, where R+ is the nonnegative part of Rn (i.e the Reihardt diagram centered at the origin).

In the last section we formulate and prove a lemma and the main theorem.

* krivokolesko@gmail.com © Siberian Federal University. All rights reserved

1. Integral representation in n-circular linearly convex domains with piecewise regular boundary in Cn

A domain G c Cn is called linearly convex ( [15], §8), if for each point z0 of its boundary dG there is a complex (n — 1)-dimensional analytic plane passing through the point z0 and disjoint from G (note that some authors use the term "weak linear convexity", for example, see [16]).

Let there be given a bounded linearly convex domain of the polyhedral form (a linearly convex polyhedron)

G = { z : gl (z,z) < 0, l = 1,...,N}

in the space Cn, where the functions gl(z,z) are twice continuously differentiable in a neighborhood of the closure of the domain.

The boundary dG of the domain G is called piecewise regular, if for every nonempty edge

SJ = = Sjl n ... n Sjk =

= { z e dG : gjl (C,C)=0,...,gjk (C,C)=0, 1 < k < n} we have the inequality dgj1 A ... A dgjk = 0, or equivalently

rang

(dgj1 dgj1 \

dZ i dZ n

Qgjk Qgjk

\dZ i . . OCnJ

k.

(1)

The orientation of the faces S1,..., SN is induced by the orientation of the boundary dG, dG =

N

|J Sn and the orientation of each face Sn, i = 1,..., N induce the orientation of every (2n — 2)-

i=i

dimensional edge Sij, dS® = |J Sij, and, taking into account the orientation, we have S= —Sji.

The orientation of the edge Sjl'"jk is given by ordering of the faces Sj1,..., Sjk, and is defined inductively.

The integral representation in bounded linearly convex domains with piecewise boundary is proved in [14]. In [1] we give a detalization of the key theorem from [14] for n-circular domains, using which we derive a series of combinatorial identities. Next we need the formulation of the theorem from [1].

1.1. Detalization of the integal representation from [14]

A domain G c Cn is n-circular if for each point z = (z1,..., zn) e G it contains all points

= (zi

•>••••> ^n

eiVn ), 0 < ^ < 2n,..., 0 < < 2n.

In case of an n-circular polyhedron the functions gl depend on modules jzj| only, i. e. such

polyhedra are as follows

G = { z : gl(\z\) < 0, l =

where the functions gl(|z|) are twice continuously differentiable in a neighborhood of the closure of this domain.

Introduce the following notation:

gl\m\ =

dgl

d|Cm| ,

d2gl

d\Zs \d\Zm \ '

(2)

ze

e

g

sm

(Vgl, ) = (Vg'K| , |C| - ïz) = £ 9m|(ICm| - ïmzm),

m=i k

^wr^z|gsimi+mm), =1 At>=i, ■■

t=l

t=l

RJk = -

0 ■ ■ ■ 0 gj ■ gl 1l ■ g i ■ glnl

0 ■ IZiIgjl ■ ■ ■ 0 ■■ IZiIgjkl gjk ■ gl 1l ■ L ii ■ Jk ■ glnl ■ Lin

IZnIg°lnl ■ ■■ IZnIgjkl -ni ■ ■ Lnn

We denote

J =

L Pi---Pk

k(k+i) (-l) 2

g

g

g

lPkl

■■■ glPkl (-i)Pi+...+Pk diz ipi

,Pk ]

X Pi---Pk ji---jk

RJk = Rh-ù =

Aï ■■■^k

^ j IZ I- ïz)* ■■■ j IZ I- Çz)*k

(3)

(4)

(5)

(6)

(7)

where AP^'j = 0 and d\Z\[p1,... ,pk] is the exterior product of the differentials d|Ci|, • • •, d\(n\, where d\Zp 1 \,..., d\ZPk \ are omitted.

Note that if 1 ^ k ^ n we find the mixed Levians Lj (gj 1,..., gjk) = LJk (gJk) from

Li(gji, ■ ■ ■, gjk )■ (8)

\j\=n-k W \C\, M - ■ - --V |C|,

Notation (2)-(8) allow us to formulate the following theorem from [1].

Theorem 2. Let G = { z : gl(\z\) < 0, l = 1, . . . , N} be a bounded piecewise regular linearly convex domain in Cn. Then every function f (z) holomorphic in G and continious on G is representable in G as

f(z) = (-i)nY,(-i)k-1E' E

,, 3 i- - - 3k

(9)

k=1

%J=k lIl=n-k

where stands for summation over ordered multi-indices Jk of length k : 1 ^ j1 < ... <

Jk=k

jk ^ N; Y1 stands for summation over multi-indices Ik = (i1,...,ik) with the property

\Ik \ =n-k

\Ik \ := ii + ... + ik = n - k,

I !

(2ni)n

Li(gji, ■ ■ ■, gjk )Lj

lsJ l

le\=i ft IZI- ïz)^1 t=i 1

f ( W»■■■» ¥ ) #

ï '

dï dïi dïn -

— = —— A ■ ■ ■ A ——, LI is a mixed Levian, and Ik! = i1! • ■ ■ ■ • ik!.

ï ïi ïn

sm

j ...jk

V

1.2. Some properties of the projection of Cn on Rn

+

For a point 2 = (zi,... ,zn) e Cn we denote \z\ = (\z1\,..., \zn\) e R+. Consider a projection of Cn on the Reinhardt diagram ( R+ )

n : (zi, ...,zn) ^ (\zi\,..., \zn\). (11)

We denote the image of the set G C Cn under this projection as C R+.

The projection n maps n-circular sets G C Cn bijectively to sets |G| = n(G) C R+.

Let

a\Z\ + ... + anzn + c = 0 (12)

be an analytic plane in the space Cn that passes through the point Z (c = — (a1Z1 + ... + anZn)) disjoint from the domain G G Cn.

The description of the projection (12) on the Reinhardt diagram is given in Proposition 4.3 [17].

Projection (12) is given by the system of inequalities:

+ |ai||zi| - |a2||z2| - |a3||z3| - ... - |a„||z„| - Ici < 0, — |ai||zi| + |a2||z2| - |a3||z3| - ... - |an||z„| - C < 0,

(13)

„—|«i||zi| — |«2||z2| — |a3||zs| — ... — |«n||zn| + C < 0.

Let us assume that an analytic plane (12) passes through the point Z i.e. c = — (a1Z1 +... + anZn). Then each plane of the family

a1z1 e-iLpi + ... + anzne-iLPn + c = 0, 0 < < 2n, m =1,...,n (14)

passes respectively through point Zeip = (Zeipi,..., ZeiPn).

It follows that the set (14) is an n-circular set in Cn. Moreover its projection on the Reinhardt diagram is given by the system (13) and coincides with the projection of each plane on the Reinhardt diagram of the family (14).

The properties of the projection n for n-circular sets give us the following geometric criterion of linear convexity of a domain G c Cn at points Z of the boundary dG.

Criterion 3. Let an analitic plane (12) be disjoint from an n-circular domain G c Cn and pass through the point Z e dG the boundary of the domain G. Then the system of inequalities (13) on the Reinhardt diagram is disjoint from the domain = n(G), |Z| belongs to the closure of the set given by the system (13), and |Z| e d|G|.

Let the set given by (13) be disjoint from the domain B c R™ and |Z| e dB belong to the closure of the set given by the system (13), then the n-circular family of planes in Cn

e-ipi + ... + a^e-^™ + c = 0, 0 < Lpm < 2n, m = 1,...,n (15)

disjoint from the domain G = n-1(B) c Cn and passing through the point |Z\evp e dG corresponds to the system (13).

Definition 1. A set G c Cn is called complete if together with the point z0 = (z0,..., zn) e G the points {z = (z1,..., zn) : 1 ^ ..., ^^ ^ ^n |} belong to G.

It follows from the definition that a complete set G c Cn is n-circular, since with each point z = (z1, ...,zn) e G it contains all the points z • eip = (z1 ■ eipi,... ,zn ■ eiPn), 0 < p1 < 2n,..., 0 < pn < 2n, and also contains the origin.

1.2.1. Some properties of n

Define in R+ the following "planes " :

+ \ai\\zi\ — \a2\\z2\ — \as\\z3\ — ... — K\\zn\ — \c\ =0,..., (ni)

-|ai||zi| — \a2\\z2\ — ... — \an-i\\zn-i\ + \an\\zn\ — \c\ = 0, (nj

— \ai\\zi\ — \a2\\z2\ — \a3\\z3\ — ... — MM + \c\ = 0, (ïï„+i) (16)

and "spaces "

+ \ai\\zi \ — \a2\\z2\ — iasiizsi — ... — \an\\zn\ — \c\ > 0,..., (+ni )

— \ai\\zi\ — \a2\\z2\ — ... — \an-i\\zn-i\ + \an\\zn\ — \c\ > 0, (+nn)

— \ai\\zi\ — \a2\\z2\ — \a3\\z3\ — ... — \an\\zn\ + \c\ > 0. (+n„+i) (17)

It is evident thNauka,at +n O +nm = 0 when l = m; l,m = l,...,n + 1 and n (13) = 0, when l = 1,...,n +1 and U ... U +nn+i U (13) = R+. Therefore

(a) if an analytic plane aizi + ... + anzn + c = 0 in Cn is disjoint from an n-circular domain G G Cn then belongs to one of the "spaces" +ni,..., +n„+i in R+;

(b) if an analytic plane aizi + ... + anzn + c = 0 in Cn is disjoint from an n-circular domain G G Cn and passes through the point Z G dG then the point \Z\ belongs to one of the "planes" nm when m = 1,... ,n + 1 which is tangent to at the point \Z\ G d\G\;

Remark 4. If n-circular domain G c Cn is linearly convex then |G| is convex in R+. The converse is false, an example is given by an open ball in Rrt.

(c) if an analytic plane a1z1 + ... + anzn + c = 0 in Cn is disjoint from a complete domain G e Cn then belongs to +nn+1 i.e. for points of a complete n-circular domain G e Cn there is an inequality:

|a1||z1| + MM + ... + ^Hzr! — c < 0, (18)

from which follows

|(a, z)| = + a2z2 + ... + a^z^ < (19)

Remark 5. Recall that we consider a bounded linearly convex domain

G = { z : gl (z,z) < 0, l = 1,...,N},

where the functions gl(z,z) are twice continuously differentiable in a neighborhood of the closure of the domain. The boundary dG of G consists of the faces

Sl = { z e G : gl(z,z)=0}, l = 1,...,N.

The analytic plane tangent to the domain G at the point Z e Sl ( [15], p. 87), i. e. which passes through the point Z e dG and is disjoint from G is given by:

l zm — Zm)=0 } . (20)

Jz : (Vglc, z — Z) = £ ^f ^ — Zm) = 4 . L m=i )

Note that the number of all analytic planes tangent to the domain G at the point Z of the edge Sji...jk q dG is described by analytic planes tangent to the faces Sjl,..., Sjk at the point Z■

From (20) we see that in case of n-circular domains analytic planes tangent to the domain G at the point Zellp G Sl, 0 ^ p < 2n are given by

z ■ (VglKl, IZ I- e-icpz) = ]T

m=i

dgl dIZm I

(IZmI - e-ilPmzm)=0

■

(21)

Applying the notation (2) we write the equations of the family analytic tangent planes (21) as:

E gl\m^mzm - ^ ^ = 0,

(22)

where \£i\ = ... = \£m\ = 1.

For the points z of a complete linearly convex domain G and \£i\ of (19), we have

g\mlïmzm

i

<

E gl\ml IZm

IïmI = l by virtue (23)

It follows from (23) that for points z of a complete linearly convex domain G and \£i \

\em\ = 1

El l/= l l A A ( 1 g\mlïmz"

g\m\\Zm I - E glm^mzm = 4 1 - ^ ^-

., _1 \ _ 1 l

= Ai (1 -(auï)), (24)

where

and

I(ai,ï)I

^ ^ alm^m

m=i

< i

Ai = J2 g mm ai

m=i

glil zi gi zi A i '■■■' A i

■

(25)

(26)

2. The main theorem

Theorem 6. For every bounded and complete linearly convex domain G with piecewise regular boundary such that all the vertices Sji"-jn are distinct there is a combinatorial identity with binomial coefficients

i = l-i)-B"i)k-i £ ' £ £ (KJ + ii^■ ■■• <lmJ +<>)'QmjkiAjt,27)

k = i Jk = k lik l =n-k mjk = s

where

^mjk (s, Jk, Ik )

IZI'(® "■■■■•w" .tkJ

\SJk

k

n

t=i

Jkl EI (Ajt)lmjtl+it+i

and

Ajt = Egj IZiI, gjl\=(gii\---,gj-;), IZ Is = IZiISi •■■■• IZnI

l = 0

lzl = (gi

ajt = (ajii> ■ ■ ■ > ajtn), mjt = (™jii> ■ ■ ■ > ajtn),

(28)

(29)

(30)

s

n

mjtl a , , mi

-jt ±± j l ' x"Jt

l = i l = i

! = I! miti1, |mJt \ = mjt i + ... + '■

jtn ■.

= 11 , mjk ! = II mjt !

(31)

(32)

i=i

t=i

and

mJk = mJi + ... + mjk = (mjii + ... + mjk U mjin + ... + mjkn), (33)

\mJk \ = mjli + ... + mjki + ... + mjin + ... + mjk n = |mJi \ + ... + |mJk\. (34)

Lemma 7. For any t = 1,... ,k, if \(ajt, £)| < 1 then in notation (29)-(34)

1

-y

L. !

k Jk!

n(1 — (ajt ,^))it+i

t=i

ft (\mjt \ + it)! X y^ t=i_^ Jk

mJk =M

mJk!

! J

(35)

\

/

where ¡i = (¡1,..., ¡n) e Z++.

Proof of lemma 7. Recall that in case |(a, £)| < 1 the following formula holds 1

(1 — (a,i))'

= 1 + (—i)(—(o4))i + (—i)(—i — 1)(—(gQ)2 +

1!

2!

(—i)(—i — 1)... (—i — (i — 1))( — (aQ)1 + = ... + l! +...

l=0

œ

Z^ l!(i_ 1)! Z^

„ l!(i — 1)! 0 mi + ...+mn=l

mi!... mn\

(aiti)mi ■ ... ■ (antn)mn =

" \m\ + i — 1)! m!(i — 1)!

£

|m|=0

(aOm, (36)

where m = (mi,..., mn), (a£)m = (ai ^^ ■... ■ (an În)mn = am Îm and \m\ = mi + ... + mn. Let \(ajt , £)\ < 1, t = 1,... ,k. Applying (36), we get

n

k / 1 œ

(1 — (ajt +' " <t!

n I à £ (aj. i)m'- 1 =

1

Ik

œ œ / k

! £ ... £ n

' |mji | = 0 |mjk |=0 \t=i

\mjt \ + it)! mjt ! V"Jt

(ajt

1

Tk!

_ £ £ ^ (j^ j ^

' |m(l | = 0 |m(, |=0 \t=i jt' /

(mji +...+mjk ) =

- y

Ik!

n(\mjt \ + it)!

t=i

k! mjk !

mjk ^0 k

! "Jk

ajkJk emjk = jk-! £

£

Il (\mjt \ + it)! X

t=i_ ™Jk

mJk!

! Jk

e.

□

m

m

e

m!

1

m j =m

Proof of theorem 6. Apply the theorem from [1] to the holomorphic function f (z) = zs = zSl

■■■ • znn, (si,---,sn) G Z+.

ny G ^ + •

On every edge SJk by virtue of (10) we then obtain

Ik ! i X ( Jk) i )si • ■■■ •(¥)sn dïiA dïn

^ JLik(gk)J J TWJ^T^Ï• A■■■A

lsjki i«I=I n (^gitl'IZI-ïz)it+i

t=1

Ik ' f LkJ i -^--£ (37)

(2ni)n / lk J k j k ï

[ \sJki |Ç| = i ïs^n Aj:+i(1 -(ajt ,ï))it+i

Due to completeness of the domain, it follows from Remark 5 and formula (25), for t = 1,... ,k that \(ajt ,£)\ < 1, where \^i\ = 1,..., \£n\ = 1. Applying (35) we obtain from (37) that

Ik ! [ 1 1 / v^ dmj! \ + ii)! • ... • (\mjk \ + ik)!mJk \ _

1 V^ I V^ (Imji I + ii)! • ■ ■ ■ • (Imjk I + 'k )!„ mjk 1 № dï

u ^ mi ajk 4 • ï

(2ni)n J k , +i Ik^ mJk! ~"Jk

|€| = 1 ïs fi Ajt+i ^>0 \mjk =m Jk

1 (ImjiI + '1)! • ■■■ • (Imjk I + ik)! amjk

k . _ Z^ mT ! Jk

t=i

mJk !

jt

n Ajt+i mjk = s

jT _

1 V"^ (Imji I + )! • ■■■ • (Imjk I + ik)! A am

k -, mj ! llajT

FT Ait+1 mjk = s Jk' t =1

t=1 jt

k / jT \ mJT

1 ^ (ImjiI + ii)! • ■■■ • (ImjkI + ik)! - (^

t=i jt

v (ImjiI + ii)! • ■■■ • (ImjkI + ik)! A ( jA k mj ! H I A-

yl Ait + 1 mjk = s Jk' t = 1 \ 3t j

H Aj;nJtl+it+1 mjk = s

Then (37) becomes

E ii)! n,J-!'ImiJ + * " n (9JzT| ) ■ (38)

k lm Mi 1 ^ mjk ! t =1

t=1 jt

,ji...jk = zs • £ (Imji I + ii)! • ■■■ . (Imjk I + ik)! f \C Ist Ik (gJk ) n(gjT )mjT ^,

mjk =s mJk ! iJkl H Aj™''t l+it+1 t=^ '

t=1

or

vii- = zs • £ (Imji I + ii)! •■■■ !(Imjk I + ik )! • Hmjk (SJk ,Ik), (39)

mJk! k

k

where

"mjk MI. )= / ^^ A (^l)"" '"Jk

| S Jk |

^ A'.mjtl+it+1 t=i

t=i

^ ^" ■■•^r (40)

isJk i i=y jt

k A|.lnJt |+it+i

i

zs

Applying (39) to (9) of theorem [1], for f (z) = zs, we obtain

n

zs = zS( —1)^{ — 1)k-i £ ' £ E V \ mji\ + ii)! TllJ. ;( \ mjk \ + ik (sJkJk).

k = i J = k ^k =n-k mJk =s

mJk !

Whence a series of identities follows

1 -(-1)nt-r1 E' E E (|mjl| +n)!m";( m |+*k^(s,JkJk).

k=1 $Jk = k \Ik\=n-k mjk = s Jk!

□

I would like to express my deep apprecation to Dr. E. Leinartas, Dr. G. Egorychev and

Dr. A. Shchuplev for their helpful advices and remarks.

I am also grateful to Dr. S. LJ. Damjanovic for the works [6,9,10] which he referred me to.

The work was supported by the grant of the Russian Federation Goverment for research under

the supervision of leading scientist at the Siberian Federal Univesity, contract 14.Y26.31.0006.

References

[1] V.P.Krivokolesko, Integral representations for linearly convex polyhedra and some combinatorial identities, J. Sib. Fed. Univ. Math. Phys., 2(2009), no. 2, 176-188 (in Russian).

[2] T.W.Chaundy, J.E Bullard, John Smith's problem, Math. Gazette, 44(1960), 253-260.

[3] V.M.Shelkovich, Algebra of distributions with a point singular support, Dokl. AN SSSR, 267(1)(1982), 53-57 (in Russian).

[4] I.Daubeacheis, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

[5] D.Zielberger, On an identity of Daubeachies, Amer. Math. Monthly, 100(1993), 487.

[6] T.H.Koornwinder, M.J. Schlosser, On an identity Chaundy and Bullard. I, Indag. Math. (N.S), 19(2008), 239-261.

[7] G.P.Egorychev, Combinatorial identity from the theory of integral representations in Cn, Irkutsk Gos. Univ. Mat., 4(4)(2011), 32-44 (in Russian).

[8] V.P.Krivokolesko, E.K.Leinartas, On identities with polynomial coefficients, Irkutsk Gos. Univ. Mat., 5(3)(2012), 56-63 (in Russian).

[9] T.H.Koornwinder, M.J.Schlosser, On an identity Chaundy and Bullard. II, More history, Indag. Math. (N.S), 24(2013), 174-180.

[10] H.Alzer, On a combiatorial sum, Indag. Math., 26(2015), 519-525.

[11] V.P.Krivokolesko, Integral representation of holomophic functions in n-circular linear convex domain with piecewise regular boudary, Mat. sistemy, Krasnoyarsk, Izd. KrasGAU, 10(2011), 95-107 (in Russian).

[12] G.P.Egorychev, Integral Representation and the Computation of Combinatorial Sums, Trans. of Math. Monogr., vol. 59, Amer. Math. Soc., Providence, RI, 1984.

[13] M.N.Davletshin, G.P.Egorychev, V.P.Krivokolesko, Calculation of multiple combinatorial sums in the theory of holomorphic functions in Cn, Advances in Applied Mathematics, 48(2012), 446-456.

[14] V.P.Krivokolesko, A.K.Tsikh, Integral representations in linearly convex polyhedra, Sib. Mat. J., 46(2005), no. 3, 579-593.

[15] L.A.Aizenberg, A.P.Yuzhakov, Integral Represetations and Residues in Multidimensional Complex Analysis, AMS, RI, Providence, 1983.

[16] M.Andersson, M.Passare, R.Sigurdsson, Complex convexity and analytic functionals, BaselBoston-Berlin, Birkhauser, (Progr. Math., 225), 2004.

[17] M.Forsberg, M.Passare, A.Tsikh, Laurent Determinants and Arrangements of Hyperplane Amoebas, Advances in Mathematics, 151(2000), 45-70.

Получение комбинаторных тождеств с полиномиальными коэффициентами с помощью интегрального представления

Вячеслав П. Кривоколеско

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

На основе интегрального представления функций, голоморфных в n-круговых линейно выпуклых полиэдрах, предложен метод получения тождеств с полиномиальными коэффициентами.

Ключевые слова: комбинаторные тождества, полиномиальные коэффициенты, интегральные представления.