On the Stieltjes constants and gamma functions with respect to alternating Hurwitz zeta functions

TL;DR

This paper explores the properties and representations of the alternating Hurwitz zeta function and related constants, introducing new series, integrals, and two novel series expansions of pi.

Contribution

It provides new integral, series, and product representations for the alternating Hurwitz zeta function and associated constants, extending classical analysis results.

Findings

New series and integral representations of the alternating Hurwitz zeta function

Two new series expansions of pi involving modified constants

Connections established between classical and modified special functions

Abstract

Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $\zeta(z)$, the generalized Stieltjes constants $\gamma_k(q)$, the Euler constant $\gamma$, Euler's gamma function $\Gamma(q)$ and the digamma function $\psi(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Euler zeta function) $$\zeta_{E}(z,q)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(n+q)^{z}},$$ the alternating zeta function $\zeta_{E}(z)$ (also known as the Dirichlet's eta function $\eta(z)$), the modified Stieltjes constants $\tilde\gamma_k(q)$, the modified Euler constant $\tilde\gamma_{0}$, the modified gamma function $\tilde\Gamma(q)$ and the modified digamma function $\tilde\psi(q)$ (also known as the Nielsen's $\beta$ function). Many new integrals, series or infinite product representations of these constants and special functions have been found. By the way, we also get two new series expansions of $\pi:$ \begin{equation*} \frac{\pi^2}{12}=\frac34-\sum_{k=1}^\infty(\zeta_E(2k+2)-1) \end{equation*} and \begin{equation*} \frac{\pi}{2}= \log2+2\sum_{k=1}^\infty\frac{(-1)^k}{k!}\tilde\gamma_k(1)\sum_{j=0}^kS(k,j)j!. \end{equation*}