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International Journal of Combinatorics
Volume 2014 (2014), Article ID 712321, 14 pages
Decomposition Formulas for Triple -Hypergeometric Functions
Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden
Received 11 November 2013; Accepted 31 December 2013; Published 15 May 2014
Academic Editor: R. Yuster
Copyright © 2014 Thomas Ernst. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the spirit of Hasanov, Srivastava, and Turaev (2006), we introduce new inverse operators together with a more general operator and find a summation formula for the last one. Based on these operators and the earlier known -analogues of the Burchnall-Chaundy operators, we find 15 symbolic operator formulas. Then, 10 expansions for the -analogues of Srivastava’s three triple hypergeometric functions in terms of -hypergeometric and -Kampé de Fériet functions are derived. These expansions readily reduce to 10 new expansions for the three triple Srivastava hypergeometric functions in terms of hypergeometric and Kampé de Fériet functions.
The concept of decomposition formulas for multiple hypergeometric functions is well known from the articles by Hasanov et al.  and Bin-Saad . This paper follows the first one by using -analogues of 15 symbolic operator formulas.
The paper is organized as follows. In Section 2, we give the basic definitions and introduce a new inverse -analogue of the Burchnall-Chaundy operators for an arbitrary number of variables, together with an explicit summation formula for one of these; the proof uses a -Lauricella function summation formula. In Section 3, we express the three triple Srivastava -hypergeometric functions in terms of these operators and -hypergeometric and -Appell functions.
Finally, we find 10 expansions for these triple -hypergeometric functions. The proofs show a certain symmetry, which is explained in detail in the last section.
Definition 1. The power function is defined by . We always assume that . Let be an arbitrary small number. We will use the following branch of the logarithm: . This defines a simply connected space in the complex plane.
The variables denote certain parameters. The variables , , , , , will denote natural numbers except for certain cases where it will be clear from the context that will denote the imaginary unit. The -shifted factorial is given by Since products of -shifted factorials occur so often, to simplify them, we will frequently use the more compact notation: The operator is defined by By (4), it follows that Assume that ; that is, and relatively prime. The operator is defined by
Furthermore, The -gamma function is given by To save space, the following notation for quotients of functions will often be used: On the ring of polynomials , we define the functions by The following notation is often used when we have long exponents: The following notation will sometimes be used:
Definition 2. Generalizing Heine’s series, we will define a -hypergeometric series by
We assume that the and contain and factors of the form or , respectively. The notation denotes a multiple summation with the indices running over all nonnegative integer values. In this connection, we put .
Definition 3. The vectors have dimensions Let Then, the generalized -Kampé de Fériet function is defined by It is assumed that there are no zero factors in the denominator. We assume that contain factors of the form , , or . In the rest of the paper, we write instead of .
Definition 4. The four -Appell functions are given by The -analogues of the Srivastava triple hypergeometric functions are
Definition 5. In the spirit of Burchnall and Chaundy , the following inverse pair of symbolic operators was introduced in ; compare Jackson . Let
We limit our considerations to the case .
Then, one has Similarly, we obtain
Definition 6. Introduce the inverse pair of operators
A general operator of the type (27) is given by
We have the following summation formula for the operator : where This follows from the summation formula where the fourth -Lauricella function is given by
The proofs of the following 15 formulas are straightforward from the definitions.
Theorem 7 (-analogues of [1, pp. 959-960]). Consider
Remark 8. We note that there are equations and , , and always follow after each other five times. The operator always follows after . The slightly more involved proofs of formulas (42) to (44) are somehow connected. The formulas are symmetrically ordered (in triples to the far right) according to the scheme , , , , . We have corrected a slight error in formula [1, p. 959 (3.6)].
We will now present 10 -analogues of the improved versions of [1, pp. 960–962]. The proofs are quite similar; in each case, we obtain a 2-, 3-, or 4-sum (denoted by the letters , , , ), followed by a three-sum , which denotes a -hypergeometric function times a -Appell function. The most difficult part in each proof is the variable transformation which takes place at the step denoted by. If we denote the variables at this stage by , , , , , , , (the old summation indices) and , , , , , , , (the new summation indices), we have to find a symmetric variable transformation with nonzero determinant. The two first transformations take the forms: Throughout, we use the following abbreviations:
Proof. Use formula (33). The computation goes as follows:
Proof. Use formula (34). The very symmetric computation goes as follows:
Proof. Use formula (36). The computation goes as follows:
Proof. Use formula (37). The computation goes as follows:
Proof. Use formula (39). The computation goes as follows:
Proof. Use formula (40). The computation goes as follows: