The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

TL;DR

This paper introduces a novel summation technique using partial fraction decomposition and Taylor expansion to derive new identities involving the Hurwitz zeta and Gamma functions, specifically for arithmetic sequences.

Contribution

It presents a new summation method combining two approaches to derive identities related to special functions for arithmetic sequences.

Findings

Derived new equalities involving Hurwitz zeta and Gamma functions

Established a summation method applicable to arithmetic sequences

Connected partial fraction decomposition with special function identities

Abstract

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.