Remark on the Hurwitz-Lerch zeta function

Junesang Choi

doi:10.1186/1687-1812-2013-70

Abstract

Various generalizations of the Hurwitz-Lerch zeta function have been actively investigated by many authors. Very recently, Srivastava presented 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. In this paper, firstly, we show that by using the Poisson summation formula, the analytic continuation of the Lerch zeta function can be explained and the functional relation for the Lerch zeta function can be obtained in a very elementary way. Secondly, we present another functional relation for the Lerch zeta function and derive the well-known functional relation for the Hurwitz zeta function from our formula by following the lines of Apostol’s argument. MSC: 11M99, 33B15, 42A24, 11M35, 11M36, 11M41, 42A16.

Highlights

Introduction and preliminaries The HurwitzLerch zeta function is defined as follows:∞ zn (z, a, s) = (a + n)s n=a ∈ C \ Z– ; s ∈ C when |z| < ; (s) > when |z| =, ( . )where C and Z– denote the set of complex numbers and the set of nonpositive integers, respectively
Srivastava [ ], motivated essentially by recent works of several authors, presented 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 HurwitzLerch zeta functions
We consider only the case |z| =, i.e., z = e πix (x ∈ R), R being the set of real numbers: e πix, a, s

Summary

Introduction

1 Introduction and preliminaries The Hurwitz-Lerch zeta function is defined as follows: This function (x, a, s) can be extended to the whole s-plane by means of the contour integral (x, a, s) = ( – s)I(x, a, s), Since I(x, a, s) is an entire function of s, equation

Results

Conclusion