A Series Representation for the Hurwitz–Lerch Zeta Function

Robert Reynolds ,A D Stauffer

doi:10.3390/axioms10040279

Abstract

We derive a new formula for the Hurwitz–Lerch zeta function in terms of the infinite sum of the incomplete gamma function. Special cases are derived in terms of fundamental constants.

Highlights

We develop a novel formulation for the Hurwitz–Lerch zeta function in terms of the infinite sum of the incomplete gamma functions given by Received: 27 September 2021
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Use Equation (9) and form a second equation by replacing m → p and take their difference followed by setting k = −2, a = eπ, m = i/2, p = −i/2 and simplify in terms of Catalan’s constant C using entry (2) in table below (64:12:7) and Equation (64:4:1)

Summary

Significance Statement

Series Representation for the Hurwitz–Lerch Zeta Function. Series representations of the Hurwitz–Lerch zeta function have been extensively studied in [1,2,3]. The Hurwitz–Lerch zeta function generalizes the Hurwitz zeta function, the polylogarithm function, and many interesting and important special functions. The purpose of this work is to contribute to the existing literature on Hurwitz–Lerch zeta function series representations by giving a formal derivation of the Hurwitz–Lerch zeta function expressed in terms of the infinite sum of the incomplete gamma function. We expect that researchers will find this new integral formula helpful in their present and future study. Every new finding on the Hurwitz–Lerch zeta function is significant due to the function’s many applications in both applied and pure mathematics. This study contains no previously published findings

Introduction

The Hurwitz–Lerch Zeta Function

Hurwitz–Lerch Zeta Function in Terms of the Contour Integral

Incomplete Gamma Function in Terms of the Contour Integral

Special Cases

Discussion