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arXiv · 2405.02854

On gamma functions with respect to the alternating Hurwitz zeta functions

Wanyi Wang·Su Hu·Min-Soo Kim
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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 $\widetilde\Gamma(x)$.

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Wanyi Wang, Su Hu, Min-Soo Kim. 2024-05-05. On gamma functions with respect to the alternating Hurwitz zeta functions. https://arxiv.org/abs/2405.02854

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