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Open Access Research

Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions

Hari M

Author Affiliations

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

SpringerPlus 2013, 2:67  doi:10.1186/2193-1801-2-67

Published: 25 February 2013

Abstract

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.

2010 Mathematics Subject Classification

Primary 11M25, 33C60; Secondary 33C05

Keywords:
Riemann; Hurwitz (or generalized) and Hurwitz-Lerch Zeta functions; Lerch Zeta function and the Polylogarithmic (or de Jonquière’s) function; General Hurwitz-Lerch Zeta function; Gauss and Kummer hypergeometric functions; Fox-Wright Ψ-function and the <a onClick="popup('http://www.springerplus.com/content/2/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.springerplus.com/content/2/1/67/mathml/M1">View MathML</a>-function; Mittag-Leffler type functions; Mellin-Barnes type integral representations and Meromorphic continuation; Generating functions and Eulerian Gamma-function and Beta-function integral representations