On gamma functions with respect to the alternating Hurwitz zeta functions

TL;DR

This paper explores a new gamma function derived from the alternating Hurwitz zeta function, establishing its properties and relations similar to classical gamma functions, including integral, recursive, and reflection formulas.

Contribution

It extends the understanding of the gamma function associated with the alternating Hurwitz zeta, providing new properties and a Lerch-type formula linking derivatives of the zeta to the new gamma function.

Findings

Derived integral and limit representations of the new gamma function

Established recursive, duplication, and reflection formulas

Proved a Lerch-type formula relating derivatives of the zeta to the gamma function

Abstract

In 2021, Hu and Kim defined a new type of gamma function $\widetilde{\Gamma}(x)$ from the alternating Hurwitz zeta function $\zeta_{E}(z,x)$, and obtained some of its properties. In this paper, we shall further investigate the function $\widetilde{\Gamma}(x)$, that is, we obtain several properties in analogy to the classical Gamma function $\Gamma(x)$, including the integral representation, the limit representation, the recursive formula, the special values, the log-convexity, the duplication and distribution formulas, and the reflection equation. Furthermore, we also prove a Lerch-type formula, which shows that the derivative of $\zeta_{E}(z,x)$ can be representative by $\widetilde\Gamma(x)$.