A Short Calculation of the Multiple Sum of Krivokolesko-Leinartas with Linear Constraints on Summation Indices Текст научной статьи по специальности «Математика»

Онлайн-доступ к журналу: http: / / mathizv.isu.ru

Серия «Математика»

2019. Т. 29. С. 22-30

УДК 519.1 MSG 15 +16

DOI https://doi.org/10.26516/1997-7670.2019.29.22

A Short Calculation of the Multiple Sum of Krivokolesko-Leinartas with Linear Constraints on Summation Indices

G. P. Egorychev

Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract. The method of integral representation and calculation of the combinatorial sums of various type (the method of coefficients) using the formal Laurent power series over C, the theory of analitical functions and the theory of multiple residues in C" were proposed by the author in the late seventies. This method was applied in various fields of mathematics. The method of coefficients is important for a difficult problem of calculation of the multiple sums with linear constraints on summation indices. Various combinatorial problems can be formulated in terms of such constraints. The calculation of the multiple sum with (/-binomial coefficients and linear recurrent constraints on summation indices was published by the author in «The Bulletin of Irkutsk State University. Series Mathematics.»in 2016. This problem appears at the enumeration of all own i-dimensional subspaees of the space Vm over field GF(q). V.P. Krivokolesko and E.K. Leinartas in «The Bulletin of Irkutsk State University. Series Mathematics.» in 2012, using the Hadamard composition have proved the multiple identity with polynomial coefficients and various constraints on the limits of summation, containing the family of free parameters. This identity is generalisation of the identities studied earlier by several authors, since constructions of the Deubechies filters in the wavelets theory. Using the author's method of coefficients the short and simple calculation of Krivokolesko-Leinartas sum is carried out. These calculations also automatically provides an equivalent way of calculation of the specified sum by means of a traditional method of generation functions, using only the well-known operations over corresponding multiple formal Laurent power series.

Keywords: combinatorial sums, the method of coefficients, integral representations, generating functions

1. Introduction

Recently, V.P. Krivokolesko and E.K. Leinartas [8] with the Hadamard composition properties proved the following statement: Let ¡j, be the given natural number, 1 ^ ¡j, ^ n, and complex numbers z = (zi,...,zn) such that

\Zi\ + . . . + l^-i | <1, Zi+Z2 + ... + Zn = l. (1.1)

Then the following multiple summation formula with polynomial coefficients and linear constraints on the summation indices holds:

Sn = £ HTTjn^-1- M

Here /3 = (/3i,..., (3n) and for j = /x, ¡j, + 1,..., n set

Bti,j = {P Pn^Sp,..., Pj-i < Sj-i,Pj = Sj,Pj+1 < Sj+i, ...,Pn<sn}

and polynomial coefficient

'Pl + ...+Pn\ (/?! + ... +Pn)\

Pl,...,Pn J Pl\...Pn\

Generalization of combinatorial identities studied in [2], [11], [10], [4] gives us formula (1.2).

Let us conduct multiple calculation (1.2) using the general scheme of the coefficient method [3], [6]. Many applications of this method in various fields of mathematics were found. Here readers may refer to [12], [9], [7], [1], [5]. First of all, let us give a brief description of the coefficient method.

2. Coefficients method

Let L be the set of Laurent power set with real or complex coefficients, containing only a finite number of terms with negative powers; Lk = {A(z)} is set of formal Laurent power series of order k,

k oo

A{z) = Y^ a-iZ+ Y aizak / 0. i=0 i= 1

For A(z) € L we define formal residue operator

resz{A(z)} := a-!. (2.1)

Two series are similar A(z) = ~}2kakzk and B(z) = ^kbkzk from L if and only if ak = bk for all k. Operator resz{A(z)} is defined on the set

L Laurent formal power series with the well-known operations of addition, multiplication, superposition and inversion, as well as the operations of differentiation and integration of series (see, for example, [3]). If

oo

A(z) = ^anzn,

n=0

then from (2.1) it follows

an = resz{A(z)z~n_1}, n = 0,1,.... For example, one of the possible representations for a binomial coefficient

Q =resz{(l + z)nz-k-1},k = 0,l,...,n,

and for polynomial coefficient

.....-fc^^n*-}. <-)

From the operator definition resz{A(z)} and series properties L the rules of action (output rules) for the operator resz{A(z)} follow directly. We will omit the replacement, inverse, differentiation and integration rules and present only those that are used by us in the calculations.

Operator output rules resz{A(z)}

Let us consider the following two series A(z) = ^fcafc-2fc and B(z) = J2k bkZk from L.

Deduction rule. Equality

resz{A(z)z~k~1} = resz{B(z)z~k~1}

is fulfilled for all к = 0,1,..., if and only if A(z) = B(z). Linearity rule. For all a, fi € С

aresz{A(z)} + f3resz{B(z)} = resz{aA(z) + (3B(z)}.

By induction, it follows that the operators resz{A(z)} and summation are commutative for any finite number of terms. Substitution rule.

YJWkresz{A{z)} = [A(z)]z=w = A(w). к

If series A(z) converges in a punctured neighborhood of the origin, then we assume

resz{A(z)} = resz=0{A(z)},

where resz=o{A(z)} is classic residue at z = 0 in the theory of functions of complex variables, if necessary, we can use theory of residues for one or more complex variables [3].

3. Main result

Let

S(w)= Y, Snws^...wsn" (3.1)

Su ■ --Sn —0

be the generating function of power type from complex variables w = (wn,... ,wn) for sequence {Sn} from (1.2) non-negative integer parameters Sf-1) ■ ■ ■ i Sn-

Theorem 1. (a) The following formula is valid:

SM = fl TTTT- (3-2)

1 - IVi

l=ß

(b) If condition (1.1) holds then identity (1.2) is fulfield.

Proof. Using the integral formula (2.2) for a polynomial coefficient, by the rule of linearity we get

<s.=x> y (M-+/")îi

zßi

n OO S/Li •••[.?] n

Er1 E E n

j=ß ßi...ß,,-i=0 ßti...\j]...ßn= 0

1 n

n

-ßi-1

(1 -Xi - ... -xn) . .

X;

n z- ( z-\Sj n 1 Sj ( z-^ßi

*•<••••••<•• S^lfAy ) II - £ (-

J=ß 1=1,1^3 ßj= 0

OO ( ^ £1—1

ßi—ßfj.-i=o K v nJ i= 1

Summing up the indices i = /x,..., n, i / j, (n — ¡jl)-times according to the formula of finite geometric progression

1 si / \ ft _L V | —

/Y> . / J \ Гр ■

kL") \ -JL I

% f3j=о 4 г

1 - {zl/xi)^+l

Xi Zi

and summation over indices f3\,..., fi^-i in square brackets according to the substitution rule (/x — l)-times: replacements Xi = Zi, i = 1,..., /x — 1. We have

1 - {zl/xl)Si+l

Sn = resx....., I ZL (zlY ]"[

' J ry . \ ry . / X X ryt . _ y.

]=,и J J/ 1=1, гф]

1

(1 - Xi - ... - xn)

Её ёг п

resXll...xn

з

3=p-

Xi = Zi, i=l,...,fl— 1 J

1 - (Zt/XtY^1

Xi Zi

h _ w • (3-3)

V-L Z\ ... Z/j,— 1 X^ . . . Xn) J

Using the integral formula (3.3) for the sum Sn, by the linearity rule we

get

S(w)= J2

WJ ... wr

Её ёГ П

3

3=P-

1 " {zl/xi)^+l

Xi Zi

(1 Z\ ... Z^i— 1 Хц ... xr,

n OO , \ a1

Zi \ - / ZiW - \ 3

ZL

j=n ] sj=0

Si n

■3W3 \

E

1 / • 1 • / • ^ ^ n

1

w/ —

Zi i ZiWi

_ \ i • (3-4)

V-L Z i ... Zfjj— 1 Xfi ... xr>

Let us find the sums by indices using the formula of infinite

geometric progression in assumption \wiZi/xi\ < 1, i = ¡i,... ,n and \wi\ < 1, i = ¡j,, ..., n, i / j. We have

1

Xi Zi

E

«¿=0

Si Zi I ZiWi

Wi---

œ ■ \ 'y ■

^г \ ^г

1

1

Xi Zi \ 1 Wi Xi ZiW%

1 00 / \ Si

1 . . , . Zj \ - I ZjWj \ 3

-, i = ¡i,...,n, i^j; — V

I nr. s -'

aw \ > " rr • • 5'") " / Jî / y \

- lo»)^ - ZiWi) Xj \ x3

Hence, in accordance with (3.4), we have

I n n

. Xj - ZjWj . f1 . (1 - Wi)(Xi - ZiWi)

] = fl J J J 1=1, ijty

1

(1 Z\ ... Z^-i X^ ... xr,

n ^ n

n (T^-y

l=H j=,u

n

(1 - Zi - ... - z^-i - x^ - ... - Xn) ^ ('.Xi - ZiWi)

In the last expression for S(w) let us use the first-order residues for each variable Xi for Xi = ZiWi, and taking into account conditions (1.1) 1 - zi - ... - z^-i = z^ + ... + zn, we have

n 1 n f-i ^

(1 -Wi) W^ - ... - wn - w^z^ - ... - wnzn

1=11

n

(1 -M>0'

t=fl

Thus, item (a) is proved, and according to (3.1) and (3.2)

S(n)= Y resx^.xls(w)l\w-^-1}\ =

Sfj,...s„=0 I i=fi

n ^ n

-Si- 1

1 - Wi 1 1

t=/i t=ii

n

Y w^.-.w^llw-^-1}^!.

t=ii

□

4. Conclusion

In this article the standard scheme of the coefficients method is employed for a new, direct and simple calculation of the sum Sn from (1.2) which does not require knowledge of the answer. This proves the effectiveness of the method of coefficients when calculating multiple sums with polynomial coefficients and linear constraints on the summation indices, containing a family of free parameters. In this case, linear constraints are determined by specifying finite and infinite multiple sums with independent summation indices. Finally, the sum of Sn in Theorem 1 means that they can easily be rewritten using the conventional method of generating functions, using only well-known operations over the corresponding Laurent multiple power series.

References

1. Davletshin M.N., Egorychev G.P., Krivokolesko V.P. New applications of the Egorychev method of coefficients of integral representations and calculation of combinatorial sums. Preprint arXiv: math./ 1506.03596vl, Jun 2015, pp. 1-64.

2. Deubechies I. Ten lectures on Wavelets. SIAM, Philadelphia, 1992, 357 p. https://doi.Org/10.1137/l.9781611970104.

3. Egorychev G.P. Integral representation and the computation of combinatorial sums. Novosibirsk, Nauka Publ., 1977, 287 p. (in Russian) (English: Transl. of Math. Monographs, AMS, vol. 59, 1984, 286 p.; 2-nd ed. in 1989).

4. Egorychev G.P. Combinatorial identity from the theory of integral representations in Cn. The Bulletin of Irkutsk State University. Series Ma,them,a,tics, 2011, vol. 4, no. 4, pp. 39-44. (in Russian)

5. Egorychev G.P. The enumeration of own i-dimentional subspases of a spase Vm over the field GF(q). The Bulletin of Irkutsk State University. Series Ma,them,a,tics, 2016, vol. 17, no. 3, pp. 12-22. (in Russian)

6. Egorychev G.P. Method of Coefficients: an algebraic characterization and recent applications. Advances in Combinatorial Math., Springer-Verlag; Proc. of the Waterloo Workshop in Computer Algebra, 2008, dedicated to the 70th birthday G. Egorychev, 2009, pp. 1-30.

7. Krattenthaler Ch. A new q-Lagrange formula and some applications. Proc. Amer. Math. Soc., 1984, vol. 90, pp. 338-344. https://doi.org/10.1090/S0002-9939-1984-0727262-6.

8. Krivokolesko V.P., Leinartas E.K. On identities with polynomial coefficients. The Bulletin of Irkutsk State University. Series Ma,them,a,tics, 2012, vol. 5, no. 3, pp. 56-62.(in Russian)

9. Leont'ev V.K. Selected problems of combinanatorial analysis. Moskow, Mosk. State Tech. Univ., 2001, 182 p. (in Russian)

10. Shelkovich V.M., Yuzhakov A.P. The structure of one class asymptotic V.K. Ivanov'sdistributions. Izv. Vuzov, Ser. Math., 1991, no. 4., pp. 70—73. (inRussian)

11. Zeilberger D. On an identity of Deubechies. Amer. Math. Monthly, 1993, vol. 100, p. 487. https://doi.org/10.2307/2324306.

12. Zeilberger D. Proof of the alternating sign matrix conjecture. arXiv: math./ 940721M, 2 July 1994, pp. 1-84.

Georgy Egorychev, Doctor of Sciences (Physics and Mathematics), Professor, Institute of Mathematics and Computer Science, Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, tel.: +79135924671 (e-mail: gegorychOmail. ru)

Received 20.07.19

Короткое вычисление кратной суммы Кривоколеско — Лейнартаса с линейными ограничениями на индексы суммирования

Г. П. Егорычев

Сибирский федеральный университет, Красноярск, Российская Федерация

Аннотация. В конце 1970-х гг. автором был разработан метод интегрального представления и вычисления комбинаторных сумм различного типа (метод коэффициентов) с использованием формальных степенных рядов Лорана над С, теории аналитических функций и теории кратных вычетов в С". С тех пор этот метод нашел многочисленные применения в различных областях математики в нашей стране и за рубежом. На наш взгляд, особенно интересно и актуально использование метода коэффициентов при решении трудной проблемы вычисления кратных сумм с линейными ограничениями на индексы суммирования. Проблемы такого типа нередко возникают на практике при решении различных комбинаторных задач. Например, в 2016 г. автором в статье, опубликованной в журнале «Известия Иркутского государственного университета. Серия Математика», была вычислена кратная сумма с д-биномиальными коэффициентами и линейными рекуррентными соотношениями на индексы суммирования, возникшая при перечислении всех собственных ¿-мерных подпространств Ут над полем

В 2012 году В. П. Кривоколеско и Е. К. Лейнартас в журнале «Известия Иркутского государственного университета. Серия Математика» доказали с использованием композиции Адамара кратное тождество с полиномиальными коэффициентами и ограничениями различного типа на пределы суммирования, содержащее семейство свободных параметров. Это тождество является обобщением тождеств, изученных ранее несколькими авторами, начиная с построения фильтров Добеши в вейвлет-теории. Здесь по стандартной схеме метода коэффициентов проведено, не зная ответа, короткое и простое вычисление кратной суммы Кривоколеско - Лейнартаса. Это вычисление также автоматически дает эквивалентный способ вычисления указанной суммы с помощью традиционного метода производящих функций, используя лишь хорошо известные операции над соответствующими кратными степенными рядами Лорана.

Ключевые слова: комбинаторные суммы, метод коэффициентов, интегральные представления, производящие функции.

Список литературы

1. Davletshin M. N., Egorychev G. P., Krivokolesko V. P. New applications of the Egorychev method of coefficient of integral representations and calculation combinatorial sums // Preprint arXiv: math./ 1506.03596vl, Jun 2015. P. 1-64.

2. Deubechies I. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992. P. XIX+357.

3. Егорычев Г. П. Интегральное представление и вычисление комбинаторных сумм. Наука : Новосибирск, 1977. (English: Transl. of Math. Monographs, 59, AMS, 1984, 286 p.; 2-nd ed. in 1989).

4. Егорычев Г. П. Комбинаторное тождество из теории интегральных представлений в С" // Изв. Иркут. гос. ун-та. Сер. Математика. 2011. Т.4, № 4. С. 39-44.

5. Егорычев Г. П. Перечисление собственных i-мерных подпространств пространств Vm над полем GF(q) // Изв. Иркут. гос. ун-та. Сер. Математика. 2016. Т.17. С. 12-22.

6. Egorychev G. P. Method of coefficients: an algebraic characterization and recent applications // Advances in Combinatorial Math. Springer-Verlag. Proc. of the Waterloo Workshop in Computer Algebra 2008, dedicated to the 70th birthday G. Egorychev. 2009. P. 1-30.

7. Krattenthaler Ch. A new q-Lagrange formula and some applications // Proc. Amer. Math. Soc. Vol. 90. 1984. P. 338-344. https://doi.org/10.1090/S0002-9939-1984-0727262-6.

8. Кривоколеско В. П., Лейнартас Е. К. О тождествах с полиномиальными коэффициентами // Изв. Иркут. гос. ун-та. Сер. Математика. 2012. Т. 5, № 3. С. 56-62.

9. Леонтьев В. К. Избранные задачи комбинаторного анализа. М. : Моск. гос. техн. ун-т, 2001. 182 с.

10. Шелкович В. М., Южаков А. П. Структура одного класса асимптотических распределений В. К. Иванова // Изв. вузов. Сер. Математика. 1991. № 4. С. 70-73.

11. Zeilberger D. On an identity of Deubechies // Amer. Math. Monthly. 1993. Vol. 100. P. 487. https://doi.org/10.2307/2324306.

12. Zeilberger D. Proof of the alternating sign matrix conjecture // arXiv: math./ 9407211vl, 2 July 1994. P. 1-84.

Георгий Петрович Егорычев, доктор физико-математических наук, профессор, Институт математики и фундаментальной информатики, Сибирский федеральный университет, Российская Федерация, 660041, г. Красноярск, пр. Свободный, 79, тел.: +7(391)9135924671 (e-mail: [email protected])

Поступила в редакцию 20.07.19