On gamma functions with respect to the alternating Hurwitz zeta functions

TL;DR

This work develops a full gamma-type theory for the generalized gamma function $\widetilde{\Gamma}(x)$ associated with the alternating Hurwitz zeta function $\zeta_E(z,x)$. It provides explicit integral, limit, and hypergeometric representations; duplication, distribution, and reflection formulas; weak log-convexity and explicit special values; and a Lerch-type formula linking $\log\widetilde{\Gamma}(x)$ to derivatives $\zeta_E'(0,x)$ and $\zeta_E'(0)$. It also analyzes the associated digamma function $\widetilde{\psi}(x)$, giving recursive and reflection properties and integral representations tied to $\zeta_E(n+1,x)$. By mirroring classical gamma/digamma theory in the alternating-Hurwitz setting, the results connect generalized constants to zeta-derivative values and enhance the toolkit for studying special values and analytic properties of $\zeta_E$.

Abstract

In 2021, Hu and Kim defined a new type of gamma function $\widetildeΓ(x)$ from the alternating Hurwitz zeta function $ζ_{E}(z,x)$, and obtained some of its properties. In this paper, we shall further investigate the function $\widetildeΓ(x)$, that is, we obtain several properties in analogy to the classical Gamma function $Γ(x)$, 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 $ζ_{E}(z,x)$ can be representative by $\widetildeΓ(x)$.

Theorems & Definitions (44)

• Lemma 2.1
• Remark 2.2
• proof : Proof of Lemma \ref{['Lemma 3.1']}
• Theorem 2.3: Integral representation
• Remark 2.4
• proof : Proof of Theorem \ref{['Theorem 1.10']}
• Corollary 2.5
• Corollary 2.6: Integral transforms
• proof
• Corollary 2.7