Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions.
Motivated essentially by recent works by several authors (see, for example, Bin-Saad [Math J Okayama Univ 49:37-52, 2007] and Katsurada [Publ Inst Math (Beograd) (Nouvelle Ser) 62(76):13-25, 1997], the main objective in this paper is to present a systematic investigation of numerous interesting properties of some families of generating functions and their partial sums which are associated with various classes of the extended Hurwitz-Lerch Zeta functions. Our main results would generalize and extend the aforementioned recent work by Bin-Saad [Math J Okayama Univ 49:37-52, 2007] (see also Katsurada [Publ Inst Math (Beograd) (Nouvelle Ser) 62(76):13-25, 1997]). We also show the hitherto unnoticed fact that the so-called τ-generalized Riemann Zeta function, which happens to be the main subject of investigation by Gupta and Kumari [Jñānābha 41:63-68, 2011]) and Saxena et al. [J Indian Acad Math 33:309-320, 2011], is simply a seemingly trivial notational variation of the familiar general Hurwitz-Lerch Zeta function Φ(z,s,a). Finally, we present a sum-integral representation formula for the general family of the extended Hurwitz-Lerch Zeta functions.
Primary 11M25, 33C60; Secondary 33C05.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4 Canada.
Hari Srivastava
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Fox-Wright Ψ-function and the [Formula: see text]-function; Mittag-Leffler type functions; Gauss and Kummer hypergeometric functions; General Hurwitz-Lerch Zeta function; Generating functions and Eulerian Gamma-function and Beta-function integral representations; Hurwitz (or generalized) and Hurwitz-Lerch Zeta functions; Lerch Zeta function and the Polylogarithmic (or de Jonquière’s) function; Mellin-Barnes type integral representations and Meromorphic continuation; Riemann