Abstract

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.

1. Introduction

The concept of decomposition formulas for multiple hypergeometric functions is well known from the articles by Hasanov et al. [1] and Bin-Saad [2]. 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.

2. Definitions

We start by defining the umbral notation [3], a mixture of Heine 1846 and [4].

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 [5], the following inverse pair of symbolic operators was introduced in [3]; compare Jackson [6]. Let Then, 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

3. Computations

The proofs of the following 15 formulas are straightforward from the definitions.

Theorem 7 (-analogues of [1, pp. 959-960]). Consider