Abstract
We aim at establishing a new bilateral type generating function associated with the I-function and a Mellin-Barnes type of contour integral. The results derived here are of general character and can yield a number of (known and new) results in the theory of generating functions.
1. Introduction
Bilinear and bilateral type generating functions are continuous functions associated with a given sequence and have useful applications in many research fields. For this reason, generating functions are very useful in analyzing discrete problems involving sequences of numbers or sequences of functions, in modern combinatorics. A number of generating functions and expansions of such other types of hypergeometric functions in one, two, and more variables have been developed by many authors (see [1–4]; for a very recent work, see also [5]). Here, we present a new bilateral generating function associated with the -function and Mellin-Barnes type of contour integral, mainly motivated by the work of Srivastava and Panda [4].
For our purpose, we begin by recalling some known functions and throughout this paper we will use the following notations.
Let and stand for the s-parameter sequence and , respectively, for an arbitrary complex number and for all integers .
The -function introduced by Fox [6, p. 408] will be represented and defined as follows: where where an empty product is interpreted as unity and ,,, and are positive numbers. L is suitable contour of Barnes type such that the poles of lie to the right of it and those of lie to the left of it. Asymptotic expansions and analytic continuations of the -function have been discussed by Braaksma [7].
The -function will be defined and represented as follows [8]: where and ,,, are integers satisfying ,, where is finite. ,,, are positive integers and ,,, are complex numbers. The -function is a generalized form of the well-known Fox -function [6]. In the sequel, the -function will be studied under the following conditions of existence: where
2. Bilateral Generating Functions for -Functions
In this section, we establish generating functions for the -function and Mellin-Barnes type of contour integral (3) and (1), respectively.
Theorem 1. Let , , , , , , , and be positive integers. Then the following bilateral generating function holds: where is an arbitrary complex number and is an integer .
Proof. For convenience, let the left-hand side of (8) be denoted by . Applying the integral representation of (3) to , then interchanging the order of summation and integration (which can be justified when the integral and the series involved are uniformly absolutely convergent), we get
Using the known formula (see [4, p. 273, (4.10)] ), namely,and the Mellin-Barnes type contour integral of the -function given by (1).
Finally, in view of (10) and (3), we get the desired assertion (8) of Theorem 1.
3. Special Cases
In this section, we consider some consequences of the main results derived in the preceding section. (i)If we put , -function reduces to Fox -function [6]. Then the main result (8) takes the following form: (ii)If we put ,,,,,,,, and, -function reduces to Wright’s generalized hypergeometric function [9, p. 33, (2.3.8)]. Then the main result (8) takes the following form: (iii)If we put ,,,,,,,, and, -function reduces to generalized hypergeometric function [9, p. 33, (2.3.11)]. Then the main result (8) takes the following form:
4. Concluding Remark
We conclude our present investigation by remarking that the results obtained here are useful in deriving numerous other generating functions involving various special functions due to presence of the -function given by (8). The -function, used in our results, is quite basic in nature. Therefore, on specializing the parameters of this function, we may obtain various other special functions such as Fox -function, Meijer’s -function, Wright’s generalized Bessel function, Wright’s generalized hypergeometric function, Mac-Robert’s -function, generalized hypergeometric function, Bessel function of first kind, modified Bessel function, Whittaker function, exponential function, and binomial function as its special cases, and, therefore, the result thus derived in this paper is general in character and likely to find certain applications in the theory of special functions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research of Jessada Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.