On gamma functions with respect to the alternating Hurwitz zeta functions

Abstract

In 2021, Hu and Kim defined a new type of gamma function $\tilde{\Gamma }(x)$ 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 $\tilde{\Gamma }(x)$, 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 $\tilde{\Gamma }(x)$.

21 pages