Abstract
AbstractMotivated by the recent investigations of several authors, in this paper, we derive several new expansion formulas involving a generalized Hurwitz-Lerch zeta function introduced and studied recently by Srivastavaet al. (Integral Transforms Spec. Funct. 22:487-506, 2011). These expansions are obtained by using some fractional calculus theorems such as the generalized Leibniz rules for the fractional derivatives and the Taylor-like expansions in terms of different functions. Several (known or new) special cases are also considered.MSC:11M25, 11M35, 26A33, 33C05, 33C60.
Highlights
The Hurwitz-Lerch zeta function (z, s, a) is defined by∞ zn (z, s, a) := (n + a)s n=a ∈ C \ Z– ; s ∈ C when |z| < ; (s) > when |z| = . ( . )The Hurwitz-Lerch zeta function contains, as its special cases, the Riemann zeta function ζ (s), the Hurwitz zeta function ζ (s, a), and the Lerch zeta function s(ξ ) defined by ∞ζ (s) := ns = (, s, ) = ζ (s, ) (s) >, n=∞ ζ (s, a) := (n + a)s = (, s, a)(s) > ; a ∈ C \ Z–and ∞ e nπiξ s(ξ ) := (n + )s = e πiξ, s
A more general family of Hurwitz-Lerch zeta functions was investigated by Lin and Srivastava [, p. , Eq ( )]
Relationships with the H-function, fractional derivatives, and analytic continuation formulas were established for the function defined in ( . ). It is worth noting the following special or limit cases of the function (i) For λ = ρ =, we find that λ(ρ,μ,σ;ν,κ)(z, s, a)
Summary
The Hurwitz-Lerch zeta function (z, s, a) is defined by (see, for example, [ , p. et seq.]; see [ ] and [ , p. et seq.]). A more general family of Hurwitz-Lerch zeta functions was investigated by Lin and Srivastava [ , p. A generalization of the above-defined Hurwitz-Lerch zeta functions (z, s, a) and ∗μ(z, s, a) was studied by Garg et al [ , p. )] (see [ – ]), in the year , considered a further generalization of the Hurwitz-Lerch zeta function, defined in the form λ(ρ,μ,σ;ν,κ)(z, s, a) :=. A multiparameter extension of the function was given, more recently, by Srivastava et al [ ] (see [ ]) The aim of this paper is to extend several interesting results obtained recently by Gaboury and Bayad [ ] and by Gaboury [ ] to the Hurwitz-Lerch zeta function (z, s, a) introduced and studied by Srivastava et al [ ]. Section is dedicated to the proofs of the main results and, Section aims to provide some (new or known) special cases
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