%0 Journal Article
%T New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications
%A H. M. Srivastava
%A S¨¦bastien Gaboury
%A Richard Tremblay
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/680850
%X We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (2011). These expansions are obtained by using some fractional calculus methods such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also given. 1. Introduction The Hurwitz-Lerch zeta function which is one of the fundamentally important higher transcendental functions is defined by (see, e.g., [1, page 121 et seq.]; see also [2] and [3, page 194 et seq.]) The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function , the Hurwitz zeta function , and the Lerch zeta function defined by respectively. The Hurwitz-Lerch zeta function is connected with other special functions of analytic number theory such as the polylogarithmic function (or de Jonqui¨¨re¡¯s function) : and the Lipschitz-Lerch zeta function (see [1, page 122, Equation 2.5£¿£¿ ]) The Hurwitz-Lerch zeta function defined in (7) can be continued meromorphically to the whole complex -plane, except for a simple pole at with its residue 1. It is well known that Motivated by the works of Goyal and Laddha [4], Lin and Srivastava [5], Garg et al. [6], and other authors, Srivastava et al. [7] (see also [8]) investigated various properties of a natural multiparameter extension and generalization of the Hurwitz-Lerch zeta function defined by (7) (see also [9]). In particular, they considered the following functions: with Here, and for the remainder of this paper, denotes the Pochhammer symbol defined, in terms of the gamma function, by it is being understood conventionally that and assumed tacitly that the -quotient exists (see, for details, [10, page 21 et seq.]). In their work, Srivastava et al. [7, page 504, Theorem 8] also proved the following relation for the function : provided that both sides of (11) exist. Definition 1. The involved in the right-hand side of (11) is the generalized Fox¡¯s -function introduced by Inayat-Hussain [11, page 4126] Here the parameters and the exponents can take noninteger values and is a Mellin-Barnes type contour starting at the point and terminating at the point with the usual indentations to separate one set of poles from the other set of poles. Buschman and Srivastava [12, page 4708] established that the sufficient conditions for the absolute convergence of the contour integral in (12) are given by and the region of absolute convergence is Note that when the -function
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