Some Examples of finding the sums of multiple series Текст научной статьи по специальности «Математика»

УДК 517.55

Some Examples of Finding the Sums of Multiple Series

Evgeniya K. Myshkina*

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

Russia

Received 06.05.2014, received in revised form 19.07.2014, accepted 26.09.2014 A method of finding residue integrals for some systems of non-algebraic equations are presented. Such integrals are connected to the power sums of roots for the system of equations. It is shown how the obtained results can be used for calculating sums of multidimensional series.

Keywords: residue integral, power sum, multiple series.

Introduction

A method based on multidimensional residue theory for the elimination n unknowns from a system of n non-linear algebraic equations (in the characteristic zero setting) was proposed by L.A. Aizenberg [1]. Its further developments were implemented in [2-4]. The algorithmic method (inspired by the Aizenberg and Yuzhakov strategy) introduced by M. Elkadi and A. Yger [5]. The basic idea of the method is to find certain residue integrals connected to the power sums of roots of a given system of equations (in the positive powers) avoiding finding the roots, and to apply then the recurrent Newton formulas. This method is less time-consuming and does not increase the multiplicity of the roots in comparison with the classical method.

The set of roots of a system of n non-algebraic equations in n variables is in general infinite. Moreover, multi Newton sums (with exponents in Nn) of the roots of such systems lead usually to divergent series. In the present work, we attach residue integrals to specific systems of n nonlinear equations, compute such residue integrals, and deduce from this computation (provided such series do converge) the values of the sums of multi-Newton series (with exponents in (— N*)n) formed with the roots of such non-linear systems which do not belong to the union of coordinate planes.

In the papers [6-10] a class of systems of equations containing entire or meromorphic functions was considered. In [11] a computer algebra algorithm that computes the corresponding residue integrals and applies to them the recurrent Newton formulas is presented.

Our goal is to generalize statements from the papers [6-10] to a another class of systems of non-algebraic equations; to obtain formulas for calculation of residue integrals, to give connection with power sums and to give the corresponding computer algebra algorithm.

In [6,7], the following system of functions was considered:

fi(z), f2(z),...,fn(z),

where z = (zi, z2,..., zn). Each fj(z) is analytic in the neighborhood of 0 G Cn and has the form

fj(z) = z^' + Qj(z), j = 1, 2,..., n,

*[email protected] © Siberian Federal University. All rights reserved

where ßj = (ej , ej, • • • ) is a multi-index with integer nonnegative coordinates, ze = zf1 •

ßj ßj ■

z22 • • • znn, and \\ßj || = ßj + ßj + • • • + ßji = kj, j = 1,2,..., n. Functions Qj are expanded in a neighborhood of zero into an absolutely and uniformly converging Taylor series of the form

Qj (z) = E

aaz , a H >kj

where a = (a 1, a2, • • •, an), aj ^ 0, aj G Z, and za = z a1 • z^2 The formulas for calculation of residue integrals

1 f 1 df

Y (r)

J (2ni)^7(r) ze+U f

in terms of coefficients of Qj(z) were obtained.

Then such systems was considered in [8,11]. One received multidinensional Newton formulas for such systems.

In the papers [9,10] was considered the class of systems in which functions

fj(z) = (zej + Qj(z))ePj, j = 1, 2,.. ., n, (1)

and in the paper [10] is given computer realization of considerable method.

Here we consider the system [12] in the case when the monomials ze in the system (1) are replaced with products of linear functions.

1. Residue integrals

We consider a system of functions fi)(z), f2(z),..., fn(z) and a system of equations

'fi(z) = [(1 - aiizi)™11 • ... • (1 - ai„z„)min + Qi(z)] ePl(z) = 0, f2(z) = [(1 - a2izi)m21 • ... • (1 - a2nz„r- + Q2(z)] = 0, (2)

fn(z) = [(1 - a„izi)mn1 • ... • (1 - a„„z„)ra™ + Qn(z)] ePn(z) = 0,

where m;j are natural numbers, aj are complex numbers, different for each fixed j, (z), and Qj(z) are entire functions.

Denote by qj(zi,..., zn) expression of the form

9j(zb...,z„) = (1 - a;izi)mi1 • ... • (1 - ajnz„)mm, i = 1,...n. (3)

Then, our system can be rewritten as

fi(zi,.. ., z„) = [q;(zi, .. ., z„) + Qj(zi,. .., z„)] ePi(zi'-'Zn), i = 1, 2,. .., n. (4)

For each i we define a function

{qi(z), если aj = 0, для всех j;

^11 n (5)

q;(z) • — • ... • —, если ajj = ... = ajjk =0. w

zji zjk

The system of equations

h;(z) = 0, i = 1, 2, ..., n (6)

z„

n

has n! isolated roots in C (C is a theory of functions space). Let J = (j1;..., jn) be multi-index is a permutation of (1,..., n). Then the roots of (6) are

= J (1/aij!,..., 1/a„jn), if all akjk = 0, k =1,...,n;

2ikjik

aj ^ (1/aiji ^[¿i],..., ^[¿k],..., l/a„j„ ), if ailjii = ... = aikji. =0,

where k, j = l,..., n.

Denote by rh the cycles

rh = {z G Cn : |hi| = ri, r > 0, « =I7n>.

For the case when all ak,jk =0 we define a cycle rh,aj by

|1 - aijizi| = ri, |1 - a2j2 Z2 | = r2,

, 11 anjn zn | rn .

If aiijil = .

= ... = aikjik = 0 for some «i,..., ik then rh aj is defined by

1 - aijizi| = ri, 1

1

(7)

(8)

(9)

1 1 anjn zn 1 rn ?

Lemma 1. For sufficiently small r a global cycle rh has connected components (local cycles) in the neighborhoods of the roots aj. Moreover, rh is homologous to the sum of the local cycles

rh,aj .

Consider the system of equations

Fi(z, t) = (qi(z) + t • Qi(z))ePi(z) =0 i = 1, 2,..., i

(10)

depending on the real parameter t > 0.

Let r1;... rn > 0 be the fixed real numbers. Then, for sufficiently small t > 0, the inequalities

hold on the cycles

ki(z)| > It • Qi(z)|, i = 1,... ,n.

rh = {z G Cn : = ri, i = 1,...,n}

because the cycles rh are compact.

By JY (t) we denote the residue integral

Jy (t)

1

1 dF

dFi dF2

i A —2 A ...A

(2^V=î)^rh zYi + i • z272+i • • • znn+i Fi F

dFw

Fn

=r

z

ii

r

z

where y = (71,... y„) is multi-index. Denote

Gj(z,t) = qi(z)+ t • Qi(z), i = 1, 2,. .., n.

Let I be a multi-index of the length n, consisting of s ones and n — s zeros (s = 0,..., n). Denote by A/ Jacobian of the system of functions such that to each "one" on the j-th place in I there corresponds j-th row of the derivatives (dGj/dzj), 1 < i < n in A/; and, to each "zero" on the k-th place in I there corresponds k-th row of the derivatives (dPk/dzj), 1 ^ i ^ n in A/.

Theorem 1 ( [12]). Under the assumptions made for the functions F defined by (10) the following formulas for JY (t) as convergent series are valid:

j (t) = EEE(-t)||a||(-1)

)s(j)_

J I as

P(as,J )!

Ai (t)

Yi + 1

zYn+1 zn

Qa (I ) (i,J )

where ( — 1)s(J) = 1, if J is even permutation, and ( — 1)s(J) = —1, if J is odd permutation, as is multi-index of order s, i; is a number of l-the unit of I, qa +/(I, J) = +1[j1] • ... • qas + 1[jn], and qp[jp] is product of all (1 — ap1z1)mp1 •... • (1 — apnzn)mpn besides (1 — ap. Zjp)mpjp, Qa (I) =

Q? •... • Q?,

^(as, J) = (m1ji • (aSi +1) — 1, .. ., mSJ„ • (a?„ + 1) — 1),

P(as,J)!^(mpjp • (j + 1) - 1)!,

p

^miji •(aS1 +1)-1+...+msjn •(j + 1)-1

miji •(aS1 + 1)-1

2. Residue integrals and power sums

Under certain restrictions on Q, and P, the considered residue integrals are connected to the power sums of roots of the system (2). Suppose that Q,(z) are polynomials:

Qi(z) = Z1 • • • z„ E C

za i = 1, 2, ...,n,

(12)

Ml^Q

where a is a multi-index, za = z^1 • ... • z^", and degzj Qj ^ mj, i, j = 1,..., n for all non-zero ajj. If ajj = 0 then there is no restriction on degzj Qj. Functions Pj (j = 1, 2,..., n) are the polynomials

Pj (z)= E j zn,

Q<lMl<Pj

where n = (n1,..., nn) is a multi-index.

1

Assuming that all wj = 0, we substitute zj = —, j = 1,..., n in the functions

(13)

Fi(z,t) = (q,(z) + t • Qj(z)) ePi(z), i = 1, 2,. .., n. Consequently, for i = 1,..., n we get

f,( -L ,...,—= (,„ ( _L ,...,-L ) +1. q, ( _L ,...,—

\W1 -n / V Uj U^/ V-1 -n

Pi ——

1

1

z=aj

msjn •(«L + 1)-1

• ... • dz„

j

And finally we arrive at

= I I 1 - aii — wi

F;( —,..., — ,t| =

1 \ mi

1

wi

1

wi

n

min

1 - ain- + t • Qi —,...,- e

wn / V wi w,

Pi -i,

wi ' ' Wn y =

— • (wi - aii)mii • ... • (wn - ain)min +

+t • Qi ( —,...,— ) ) eP^wii "J =

vwi wn y y

1 \ min , i i ^

— ) • (Vi(w)+ t • Qi(w)) ePHWT

wn

(14)

where qi are the functions

qi = (wi - aii)mii • ... • (wn - ain)min,

and Qi are the polynomials

Qqi = wm

Qi( —,...,— ).

Wi w„

In the above calculations it is not important whether aij vanish or not. Indeed, assume that in Fi(z, t) = (q^z) +1 • Q^z))eP(z), i = 1,..., n, some aij = 0 vanishes. If, for instance, aii = 0,

then after substitution z, = —, j = 1,..., n, the function Fi takes the form

wj

1

1

F-i ( —,...,—,t

wi wn

qi ( —,..., — ) +1• Qi( —,..., —

wi wj Vwi wn

Pi —wL

o i ^ wi ' ' wn y

i = 1, 2,. . .n.

Then

Fi —,...,—, t = 1 - ai2 —

>i wn y VV wi

1 - ain— +

wn

+t • Qi — ,...,— e

w-

wn min

Pi —,••• -1

- i V wi ' wn J =

/Y 1 ^ 1 "

(wi)degwi Qi • ... • (wn - ain)min +

+t • QA —,..., — ) ) eM ^-•• ^J =

Wi wn

1

— ) • (qi(w) + t • Qi(w) ) ePil W

W1 ' Wn

1 \ deSw! Q1

wi y "

where q is the function q = (w1)degwi Ql • ... • (wn — a1n)min, and (Q1 are polynomials of the

degwi Qi

form <5i = wi

From (12) we derive that

1

cin • QH —,... — . That is, one can take mii = deg Qi. Vwi wn ' i

degw, <31 < mij, j = 1 . . . ,n.

mii

mii

n

m

w

n

mi2

Denote

G,^, t) = qi(ui) + t • Q,^), i = 1, 2, .. ., n.

(15)

When 0 ^ t ^ 1, the system (15) has a finite number of roots in Cn, depending on parameter t, and has no infinite roots in Cn (see [13]). Sufficiently close to zero t on the cycle

qh = {- G Cn :

compactness of the cycle implies

m -......

- 1 - n

= £,, i = 1, 2,..., n},

q,

1

1

>

t • Q, ( — ,...—

- 1 - n

i = 1, 2,...,n.

- 1 - n

Therefore qh is homologous to the sum of the cycles qh,aj

1 - a1ii Wi 1 - a2,2 W"

1 ani~

= £1,

= £2,

£n .

obtained from the cycles rhjOJ by the substitution z, = —.

The equation

1 - aj,,- —

= £

defines a circle. Indeed, let us first rewrite it in the form

1 ajij -j

£, then - - aj,. | = £ |-j |

Thus

l-j - ajj |2 = £2|-j |2, then (1 - £2)

1 £2

22

£a

1

(1 - £2

1 - £2

2 £2 •lajj, |2

(1 - £2)2

j = 1,...,n.

(16)

For sufficiently small £ the point ajj lies inside this circle, and therefore rh,aj is homologous to the cycle

|w — a1j11 = £1, |w2 — a2j2 | = £2,

. |-n anjn 1 £n.

Here some a,-, can vanish.

Lemma 2 ( [12]). Let Pj be defined by (13), and the inequality

l1 + ... + ln < Y

j

2

TOo —

j

-1* —

j

holds for a multi-index 7 = (71,..., Yn), where lj = (lj,..., lj) and lj is a degree of P» in Zj for i, j = 1,..., n (i.e. n scalar inequalities l1 + ... + ^ 7» hold).

Then

Jy (t) =

(—1)n

Yl + 1 „„Y2 + 1 „„Yn + 1 A dG2

w^ • w.

(Inequality (17) means that it holds coordinatewise).

- , dGn

AW1 A ... A G1 G2 Gj

(18)

Lemma 3 ( [12]). Let A = A(w,t) be the Jacobian of the system G?1(w,t),...,Gn(w,t) U3 (15). Then

Jy (t)^(—t)|K||+^ ( —1)

Ke» J

where QK = Q^1 • ... • Qj", and

(J).

1

P(K,J)! dw^

A- wY1 + 1

,,Yn+1

q

K

qK+J (J)

K = {K = (k1,..., kn): there exists i such that ||K|| <7» + 2, i = 1,..., n}. All the notations here are as in Theorem 1.

Denote by z(j)(t) = (zj1(t),..., zjn(t)), j = 1, ...,p the zeros of the system (2) with the functions tQj, where Q» are defined by (12) and do not lie on coordinate subspaces. Since wj

do not lie on coordinate subspaces, then zjm =-, m = 1,..., n and therefore we have finite

wjm

number of zeros. Consequently p ^ s.

Theorem 2 ( [12]). The following equality holds:

p

E

1

=1 Zj1(t)Yl + 1 • Zj2 (t)Y2 + 1 ••• Zjn(t)

Yn + 1

j=1

]T(—t)||K||+n£ (—1)

s(J)_

Ke»

P (K,J)! dw^

A(t) • wY

• wY1 + 1 • • wY" + 1 .

Qq

K

qK+i (J)

Thus, the power sum of (zeros of (15)) is a polynomial on t, and therefore, the equality in

Theorem 2 also holds for t = 1.

p 1

Denote CTy+/ = E -Yl+1 _ Y2 + 1-r^^PT , where Z(j) = . . . , Zjn) = (zj1(1) . . . , Zjn (1)),

j = 1,...,n.

ZYl +1 ZY2p1 ZYn + 1' j = 1 Zj1 ^ Zj2 ^ ^ ^ Zjn

Theorem 3 ( [12]). For the system (2) with functions fj defined by (4) and Q» defined by (12) the following formulas are valid:

aY+i = E"

1

j=1 Zj1 • zj2 E (—1)"K"+nE(—1)s(J)

Y1 + 1 ^Y2 + 1 ZYn+1 ' • •Zjn

A • wY1+1 ...wY"+1 • ^1+1

Qk1 • ... • dz

i|kii>o

qr * • ... • qjkn + 1

(—1)||K||+n (—1)

s(J)_

Ke»

P(K, J)! dw^

A- wY1+1 • ... • wY"+1 •

Qq

K

qK+1 (J)

where z(j) = z(j)(1).

n

n

w=aj

1

n

w=aj

j

1

w=aj

(20)

3. Examples

Example 1.1. Consider the system of equations in two complex variables f/1(z1, Z2) = (1 + a1Z1 — a2Z2)e(C1Z1+C2Z2) = 0,

\/2(z1,Z2) = (1 — M1 + 62z2)e(d1Z1 + d2Z2) =0. ( )

Jacobian A = a1b2 — a2b1 different from zero.

The root of system (19) is z1 = — a2 A 62, z2 = — a1 A 61. Here we suppose, that the root not lie on the coordinate planes. Therefore a1 + b1 = 0, a2 + 62 = 0, then

( —1)71+72 • A71+72+2

a7+/ = (a1 + b1)Y2+1(a2 + b2)Y1+1.

In particular,

= A4

a(2'2) = (a1 + b1)2(a2 + b2)2 .

We make the change of variables z1 = — h z2 = —. System will go into

{/"1 = W1W2 + a1W2 — a2W1 = (w1 + a1)(w2 — a2) + a1a2 = 0, /"2 = W1W2 — 61W2 + 62^1 = (w1 — &1)(w2 + 62) + 61 62 = 0,

its Jacobian is A = (w2 — a2)(w1 — 61) — (w1 + a1)(w2 + 62). Now Theorem 3 implies

JY = CTY+/ =

= i_wi1+1 • wY2+1 • (a^)*1 (6162)k2 • A__(21)

(2ni)2^ (W1 + a1)k1+1 • (W2 — a2)k1+1 • K — 60k2+1 • (w + 62)k2+1 1 2'

" q,a j

where K = {y| 3i : Yj + 2 > k1 + k2, i = 1, 2}, a rq,aj are cycles of the form {|w1 + a1| = r11, |w2 + 62| = r22}, taken with positive orientation and {|w2 — a2| = r12, |w1 — 61| = r21}, taken with negative orientation.

Calculate these integrals, in particular, we have

7 2,2 , 2,2 2a1a26? 2a2a2b2 2a?6162; 2a26262

J(1,1) = a162 + a261--—r---—r---—T---—r~+

a1 + 61 a2 + 62 a1 + 61 a2 + 62

2a16162 2a2b2b2 2a1a261 2a^a2b2 2a2a26162 2a1a26162 + (a1 + 61)2 + (a2 + 62)2 + (a1 + 61)2 + (a2 + 62)2 + (a1 + 61)2 + (a2 + 62)2 . Therefore

A4 2,2 o 2 2a1a2b1 2a1a262 2a26162 2a26262 . .

--—T.-—2 = a2b2 + a2b2------L--2-1T2 + (22)

(a1 + 61)2(a2 + 62)2 12 21 a1 + 61 a2 + 62 a1 + 61 a2 + 62

2a262b2 2a26262 2a1a261 2a1a262 2a2a26162 2a1a26162 + (a1 + 61)2 + (a2 + 62)2 + (a1 + 61)2 + (a2 + 62 )2 + (a1 + 61)2 + (OTT^. Example 1.2. Recall the expansion of r-function an infinite product:

.. oo

e n (1 " 1 e k •

r(1 — z) fcA=A/ k

where 7 is Euler constant.

Consider the system of equations

(,, , eY(-a1Z1+a2Z2) ~ ( — a1Z1 + a2Z2 \

l/1(Z1,Z2) = —-1-T-vT^n 1--1- e k =0,

J r(1 — ( —a1Z1 + a2Z2)) fc=A k / (23)

f2(Z1,Z2) = , eY(b1 Z1-b2Z2) , = n f1 — e= 0.

JK ' r(1 — (61Z1 — 62Z2)) s J

Each function is expanded into an infinite product of functions from the system of type (20). The roots of the system (23) are the points

a2s + 62k a1s + 61k

a1 b2 — a2 b1 a1 b2 — a2 b1

In our case a162 = a2 61. Therefore

(2 2) = t = ^ (a162 — a261)4

2) = J(1,1) = ^ (a1 s + 61k)2(a2s + 62k)2 .

This series converges when ^ = and ^ = 7~.

s2 61 s2 62

Thus

a(2, 2) = J(1,1) =

En a2 62 + a2b2 ■n 2a2a262 ■n 2a1 a2b1 2a2 6162

k2s2 ^ k2s(a2s + 62k) ^ k2s(a1s + 61k) ^ ks2(a2s + 62k)

fc,s = 1 fc,s = 1 £,3-1 fc,s = 1

_ ^ 2a26162 + ^ 2a26262 + ^ 2a16262 + ^ 2^ + ^ ks2(a1s + 61k) + s2(a2s + 62k)2 +s2(a1s + 61 k)2 + ^ k2(a2s + 62k)2 +

k,3- 1 k,3- 1 k,3- 1 k,3- 1

En 2a1 a261 2a1a26162 ^^ 2a1a26262 k2(as + 6-, k)2 + + 6„k)2 +

k2(a1s + 61k)2 ks(a2s + 62k)2 ks(a1s + 61k)2

Therefore from (22) we have, that

En (aj62 — a261)4 = ^ a262 + a261 _

n4(a1s + 61k)2(a2 s + 62k)2 = ^ k2s2

fc,s-1 fc,s—1

En 2a2a262 ■n 2a1a262 ■n 2a26162 2a2 6162

k2s(a2s + 62k) ^ k2s(a1 s + 61k) ^ ks2(a2s + 62k) ^ ks2(a1s + 61k) +

fcJs-1 £,3-1 £,3-1 £,3-1

+ ^ 2a26262 + ^ 2a26262 + ^ 2a1a262 + ^ 2a2a262 +

^ s2(a2s + 62k)2 ^ s2(a1s + 61k)2 ^ k2(a2s + 62k)2 ^ k2(a1 s + 61k)2

£,3-1

2a1a26162 ^ 2a1a26262

+ ^ ks(a2s + 62k)2 + ^ ks(a1s + 61k)2 .

k,3- 1 k,3- 1

Use the identity [14, Ch. 5, Item 5.1. no. 2,12]

E (k + a)n = (n——))!^(n-1)(a),

k-0

where t(t) We obtain

E

k,s = 1

r'(t) r(t).

E

k=1

k(kn + m) m

^m+ 1) +c

oo 1 oo 1 oo 1

53 k2 (s + ak)2 = — 53 02k4 + 53 k2 ^'(ak)^

k,s=1 V 7 k=1 k=1

_1_= y_L

s2k(ak + 6s) S=1 6s3

+ H + C

oo C oo 1

6s3 +53 6s3

1

6s a

t — +7T a 6s

53 ks(k + as)2 as2k(k + as) ^ as2(k + as)2 f^ a2s3 [t(as + 1) + C] +

k,s=1

k,s = 1

o 1 o 1

+ E ^ — E 772 t(as).

Transform the expression

E

(a162 — a261)4

t n4(a1s + 61k)2(a2s + 62k)2 (a162 + a261^1 k2s2

k ,S—1 k, S — 1

(a262 + a262) E ^+

+ (40^ + 40^ — 8a1a2 — ^62^ £ J_ — (2a2a262 + 2a1 a261) £ J3 —

k=1 k k=1 k

a1

a2

k=1

—2a2^£ — 2a1a26^

+

k=1

k=1

o t if"1) o iT)

+(2a16162 — 2a26262) E 3 + (2a26162 — 2a16162) E a1

+

+

k=1

k=1

o Wo t' f ^

+(2a262 — 2a1a26162^-+ (2a262 — 2a1a26162^ V2 1 1

Consider the expression

Differentiate its by t. We have

S=1

S=1

^ tM

j3 .

k=1

t(tk)A

\ t'(tJ)

= j2 . \k=1 /t k=1

Therefore, our double series expressed in terms of one-dimensional series of the same type.

1

1

(24)

Example 2.1. Consider the system of equations in three complex variables

/1(Z1, Z2, Z3) = 1 — a1Z1 — a2Z2 — a3Z3 + a^Z1Z2 + a^z^ + a2a3Z2Z3 = = (1 — a1Z1)(1 — a2Z2)(1 — a3 Z3) + a1a2a3Z1Z2Z3 = 0,

/2 (Z1, Z2, Z3) = 1 — 61Z1 — 62Z2 — 63Z3 + 6162Z1Z2 + 6163Z1Z3 + 6263Z2Z3 = = (1 — 61Z1 )(1 — 62 Z2 )(1 — 63Z3) + 616263Z1Z2Z3 = 0,

/3(Z1, Z2, Z3) = 1 — C1Z1 — C2Z2 — C3Z3 + C1C2Z1Z2 + C1C3Z1Z3 + C2C3Z2Z3 = „= (1 — C1Z1)(1 — C2Z2)(1 — C3Z3) + C1C2C3Z1Z2Z3 = 0.

The roots of system (24) are (zj1; Zj2, Zj3), j = 1,2,3.

We make the change of variables z1 = —, z2 = — and z3 = —. Our system transforms

W1 W2 W3

into

/"1 = W1W2 W3 — a1W2 W3 — a2W1 W3 — a3W1 »2 + a1a2»>3 + a1a3W2 + a2 a3»1 = = (w1 — a1)(w2 — a2) (W3 — a3) + a1a2a3 = 0,

^ /2 = W1W2W3 — 61W2W3 — 62W1W3 — 63W1W2 + 6162W3 + 6163W2 + 6263W1 = (25)

= (w1 — 61) (w2 — 62) (W3 — 63) + 616263 = 0,

/"3 = W1W2 W3 — C1W2 W3 — C2W1W3 — C3W1W2 + C1C2W3 + C1C3W2 + C2C3W1 =

= (w1 — C1)(w2 — C2)(w3 — C3) + C1C2C3 = 0, where A is Jacobian of system (25)

A = (w2 — a2)(w3 — a3)[(w1 — 61X^3 — 63)^1 — C1)(w2 — C2) — (w1 —61 )(w2 — 62)(w1 — C1)(w3 — C3)] — — (w1 — a1)(w3 — a3)[(w2 — 62XW3 — 63)^1 — C1)(w2 — C2) — (»1 — 61X^2 — 62 )(»2 — C2)(w3 — C3)] + +(w1 — a1)(»2 — a2 )[(w2 — 62) (»3 — 63X^1 — C1)(»3 — C3) — (»1 — 61X^3 — 63 )(»2 — C2)(»3 — C3)].

Now Theorem 3 implies

= = y^ 1 i' W1W2W3 • (a1a2a3)k1 (616263X2(C1C2C3X3 • A

fc1+fc2 + fc3<^ y "

A d»2 A d»3

(w1 — 61)k2 + 1 (W2 — 62)fc2 + 1(w3 — 63)k2 + 1 • (W1 — C1)fc3 + 1(w2 — C2 )k3 + 1 (w3 — C3)fc3 + 1 '

where f^ are cycles of the form {|w — a1| = rn, |w — 621 = r22, |»3 — C3I = ^3}; {|w — a3| = r13, |»1 — 611 = r21, |w2 — C21 = r32}; {|w2 — a2| = r12, |w3 — 631 = r23, |»1 — C11 = r31} taken with positive orientation and {|w1 — a1| = r11, |w3 — 631 = r23, |w2 — C2| = r32}; {|w2 — a21 = r12, |w1 — 611 = r21, |w3 — C3I = r33}; {|»3 — a3| = r13, |»2 — 621 = r22, |»1 — C11 = r31} taken with negative orientation.

Calculate these integrals. We obtain

J(0,0,0) = a(1,1,1) = a162C3 + a163C2 + a261C3 + a263C1 + a361C2 + a362C1 +

a3C1C2C3 a3 — C3 a2616263

a2 — 62

+

61 + 62

61 — C1 62 — C2 C3 + C1

C3 — 63 C1 — 61

a1616263

a1 — 61 a36162 63 a3 — 63

C3 + C2

C3 — 63 C2 — 62 C2 + C1

C2 — 62 C1 — 61

+ +

a1 C1

a1 — C1

62C2C3 + 63C2C3 + a2a3 62 + a2a363

+

a2C2 a2 — C2

62 — C2 63 — C3 a2 — 62 a3 — 63 61C1C3 + 63C1C3 + a1a363 + a1a361

+

61 — C1 63 — C3 a3 — 63 a1 — 61

Example 2.2. Consider the system of equations

/2(z1,z2,z3) =

sina1z1 + a2z2 + a3z3 — a1a2z1z2 — a1a3z1z3 — a3a3z2z3

A/O1Z1 + a2 Z2 + a3z3 — a1a2 Z1Z2 — a1a3z1z3 — a3a3z2z3 :1 + a2z2 + a3z3 — 0102Z1Z2 — 0103Z1 Z3 — 0303Z2Z^ = 0

k2n2 J '

sin a/67z1^6363^3

a/61Z1 + 62 z2 + 63 z3 — 6162z1z2 — 6163z1z3 — 6363 z2z3 . 61z1 + 62z2 + 63z3 — 6162z1 z2 — 6163z1z3 — 6363z2z3\ _

n I1 s2n j=0,

/3(z1,z2,z3)

sinC1z1 + C2 z2 + C3z3 — C1C2z1 z2 — C1C3z1z3 — C3 C3z2z3

(26)

(27)

C1z1 + C2 z2 + C3z3 — C1C2z1z2 — C1C3z1z3 — C3C3z2z3 o {-! C1z1 + C2z2 + C3z3 — C1 C2z1z2 — C1C3z1z3 — C3C3z2z3\ _

. JJA1 mn ;=0.

Each function is expanded into an infinite product of functions from the sytems of the type (25). Transform Formula (26). We obtain

J(

(0,0,0)

E°° a162C3 + a163C2 + a261C3 + a263C1 + a3 61C2 + a362C1 +

c , 2 2 2 r

k,s,m=1

n6k2s2m2

a3C1C2C3

k,s,m=1

n6m2(a3m2 — C3 k2)

+

a1616263

k,s,m=1

+

k,s,m=1

+

k,s,m=1

+

n6s2(a1s2 — 61k2)

a26162 63 n6s2(a2s2 — 62k2)

Q36162 63 n6s2(a3s2 — 63k2)

a1 C1

k,s,m=1

n6 (a1m2 — C1k2)

61

+

62

61m2 — C1 s2 ' 62m2 — C2s2

C2 s2 — 62m2 C1s2 — 61m2 62 C2 C3 , 63C2C3

+

+

+

+

+

+

k,s,m=1

a1 C1

n6(a1m2 — C1k2)

2(62m2 — C2 s2) m2(63m2 — C3s2)

a2a362 a2a363

+

+

k2(a2s2 — 62k2) k2(a3s2 — 63k2)

+

+ E

a2C2

k,s,m=1

n6(a2m2 — C2&2)

biC1C3

+

63C1C3

■'2(6im2 — ci s2 ) m2(&3m2 — C3 s2 )

+

+ E

02 C2

k,s,m=1

f6(a2m2 — C2k2)

010363

+

010361

Second member of identity has the form Y1

k2(a3 s2 — 63k2) k2(a1s2 — 61k2) 1

fc,s,m=1 n6s2(as2 — 6k2 )(cs2 — dm2)

Use identity [14, Ch. 5, Item 5.1.25, no. 4] (if a > 0) ]T

1

we have

fc=0

(k2 + a2) 2a2

1 n cth(na)

+-^—^. Then

2a

E

n6s2(as2 — 6k2)(cs2 — dm2) ' n66ds2

E;

— 1 + n cth(n^—a/6s)

2(—a/6)s2 ^v/—a76s

— 1 + n cth(n^/—c/ds)

2(—c/d)s2 ^vZ—c^s

Eœ 1 cth(^^—c/ds) ^^ cth(^^—a/6s) cth(n^/—a/6s) • cth(n^/—c/ds)

4n6acs6 + /W5. /„J „ 05 + /l-n-5^ /__„„5 +

^ 4n6acs6 ^ 4nV—cdas5 S=1 4n5V—a6cs5 s=1

4nV a6cds4

Let 2$1(e2i,e2i; e4t,x) be a basic hypergeometric series (see,for example, [14, p. 793]). Consider known formula [14, Ch. 5, Item 5.2.18, no. 13]

__,-y»S 1 1 __,-y»S 1 '•y 1

V _ = 1 V —_ = 1 • x -2 $1 (e2i e2i- e4i x) = _1_2$1 (e2i e2i- e4i x)

Z^ e2ts _ 1 xZ^e2is_ 1 x e4t _ ! 2 $1(e , e ; e ,X) e4t _ 12$1(e ,e ; e ,X).

e2ts _ 1 x ' e2ts — 1 x e4t — 1

S=1 S = 1

Therefore

œ .1 /, \ œ 1 œ

y- cth(ts)

A-' s5 «5 + e5(e2is

S = 1

s° ^^ s° (e2ts — 1)

S= 1 S = 1

œ 1 1 /"1 1 py ^ rx ^ r cv

= Y" — + 2-^-- - dy / - dx / - dw / _0w - dW 2$1(e2t,e2t; e4t,u) du.

__-, s e — 1 70 y Jo x ./0 w J v Jo

s = 1

Integrating by parts, easy to show, that

/* 1 1 /*y 1 /* x 1 /* w 1 /* v 1 /* 1

1 dy / 1 dx / — dw 1 dv / 2$1(e2i,e2i; e4i,w) du = 1 ln4 y • 2$1(e2i,e2i; e4t,y) dy. ./0 y ./0 x ./0 w ./0 v ./0 ^ J0

Therefore

E

cth(ts)

Z(5) + 2(e4t1— 1) I ln4 y • 2$1 (e2t, e2t; e4t, y) dy.

Simplify the expression

^ cth(as) • cth(6s) = ^ 1 1 + e-2as 1 + e-2bs s4 = s4 1 _

s=1

œ 1

E si

s=1

22 1 + ^-7 + ^-7 +

s4 1 — e-2as 1 — e-2bs 4

e2as — 1 e2bs — 1 (e2as — 1)(e2bs — 1)

1

1

x

x

5

s

Now consider the expression

44

4a —t;-,,, „,-—. Substituting — = 2, we have

(e2as — 1)(e26s — 1) 6 6

1

11

+

(e2«s - 1)(e2bs - 1) (e2bs - 1)2(e2bs + 1) 2(e2bs - 1)2 4(e2bs - 1) 4(e2bs + 1)'

We note that

Thus

Therefore,

d

e2bs — 1

2bs

2s

2s

(e2bs - 1)2 (e2bs - 1) (e2bs - 1)2 "

2se

1 1 d

(e2bs - 1)2 e2bs - 1 2s db

e2bs — 1

d

œ 1 1

v— 1

Z.^ 2 «5

,s = 1

2s5 e2bs — 1

11

+

«4(e26s - 1)2 s4(e2bs - 1)'

Of the formulas obtained above the sum (28) is equal to - L ln4 y • — T2$1(e26, e26; e46,y)l dy.

8 d6

Therefore

E

k,s,m=1

n6s2(as2 - bk2)(cs2 - dm2) 3780ac

+

+

1

+

4n5 - cda 1

Z (5) +

Z (5) +

1

2(e4^v^-C/d - 1) 1

4n5%/ - abc y 2(e4^-a/b - 1)

J^ ln4 y • 2$1 (e^v7-^, e2n v^; e^v7-^, y) dy j + Q ln4 y • 2$1(e2^v/^, e^v^, y) dy j +

+

1

360\/a6cd

1

1

1 _ _ _

ln3 y • 2$1 (e2^V^, e2^V^; e4^V^, y) dy

Q

+

4n4Vabcd (e4^- 1)

---- f1 ln3 y • 2$1(e2nV^, e2-^^; e^V-^, y) dy+

4n4Vabcd (e4^-c/d - 1) ./q

3 1 f 1„3 * / 2wA/-c/d 2nA/-c7d. N J

oo 4 • „ ^^- ln y • 2$1(e V 7 ,e v 7 ;e v 7 ,y) dy-

32n4Vabcd (e4nV -c/d - 1) ./Q

1

ln4 y

d

8n4\/ abcd 7q -c/d

11

- 1

2$1(e2^-c/d, e2nV-c/d; e4nV-c/d, y)

dy-

32nV abcd 64nV abcd ad

substituting given W — = 2.

bc

f ln3 y • 2$1(e2^v^c7d,-1; -e2^v^c7d,y) dy,

Q

Thus we have, that ^(1,1,1) calculated in terms of well-known expression.

This work was supported by the Russian Foundation for Basic Research, 12-01-00007-a.

1

1

1

1

1

1

1

References

[1] L.A.Aizenberg, On a formula of the gereralized multidimensional logarithmic residue and the solution of system of nonlinear equations, Sov. Math. Doc., 18(1977), 691-695.

[2] L.A.Aizenberg, A.P.Yuzhakov, Integral representations and residues in multidimensional complex analysis, Amer. Math. Soc., Providence, RI, 1983.

[3] A.K.Tsikh, Multidimensional residues and their applications. Translations of Mathematical Monographs, Amer. Math. Soc., Providence, RI, 1992.

[4] V.Bykov, A.Kytmanov, M.Lazman, M.Passare (ed), Elimination Methods in Polynomial Computer Algebra, Math. and Appl., v. 448, Kluwer Acad. Publ., Dordreht, Boston, London, 1998.

[5] M.Elkadi, A.Yger, Residue calculus and applications, Res. Inst. Math. Sci., 43(2007), no. 1, 55-73.

[6] A.M.Kytmanov, Z.E.Potapova, Formulas for determining power sums of roots of systems of meromorphic functions, Izvestiya VUZov Matematika, 49(2005), no. 8, 36-45 (in Russian).

[7] V.I.Bykov, A.M.Kytmanov, S.G.Myslivets, Power sums of nonlinear systems of equations. Docl. Math., 76(2007), no.2, 641-645.

[8] A.A.Kytmanov, Analogs of Recurrent Newton Formulas, Russ. Math., 53(2009), no. 10, 34-44.

[9] A.M.Kytmanov, E.K.Myshkina, 2013, Evaluation of power sums of roots for systems of non-algebraic equations in Cn, Russ. Math., 57(2013), no. 12, 31-43.

[10] A.A.Kytmanov, A.M.Kytmanov, E.K.Myshkina, 2014, Finding Residue Integrals for Systems of Non-algebraic Equations in Cn, J. of Symbolic Computations, 66(2015), 98-110.

[11] A.A.Kytmanov, An Algorithm for Calculating Power Sums of Roots for a Class of Systems of Nonlinear Equations, Programming and Computer software, 36(2010), no. 2, 103-110.

[12] A.M.Kytmanov, A.A.Kytmanov, E.K.Myshkina, Finding and Computation Residue Integrals and Power Sums for the Systems of Non-Algebraic Equations in Cn, J. of Symbolic Computations (to appear).

[13] A.K.Tsikh, The Bezout theorem in the space of theory functions. On the solution of a system of algebraic equations, Nekotorye voprosy mnogomernogo kompleksnogo analiza, Inst. Phys. SO AN SSSR, Krasnoyarsk, 1980, 185-196 (in Russian).

[14] A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev, Integrals and series. Vol. 1. Elementary functions, Translated from the Russian and with a preface by N.M.Queen, Gordon & Breach Science Publishers, New York, 1986.

Некоторые примеры нахождения сумм кратных рядов

Евгения К. Мышкина

Рассмотрен метод нахождения вычетных интегралов для определенных систем неалгебраических уравнений. Такие интегралы связаны со степенными суммами корней системы уравнений.

Показано, как эти результаты можно применить к нахождению сумм кратных рядов.

Ключевые слова: вычетный интеграл, степенная сумма, кратные ряды.