Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives

TL;DR

This paper derives asymptotic expansions, bounds, and series representations for the alternating Hurwitz zeta function and its derivatives as the parameter grows large, advancing understanding of its behavior and properties.

Contribution

It provides the first comprehensive asymptotic expansion and bounds for the alternating Hurwitz zeta function and its derivatives, including new series representations.

Findings

Asymptotic expansion of $ ext{zeta}_E(s,q)$ as $|q| o obreak\nobreak ightarrow ext{infinity}$.

Asymptotic expansions for derivatives of $ ext{zeta}_E(s,q)$ with respect to $s$.

New exact series representation of $ ext{zeta}_E(s,q)$.

Abstract

Let $$ \zeta_E(s,q)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+q)^{s}} $$ be the alternating Hurwitz (or Hurwitz-type Euler) zeta function. In this paper, we obtain the following asymptotic expansion of $\zeta_{E}(s,q)$ $$ \zeta_E(s,q)\sim\frac12 q^{-s}+\frac14sq^{-s-1}-\frac12q^{-s}\sum_{k=1}^\infty\frac{E_{2k+1}(0)}{(2k+1)!}\frac{(s)_{2k+1}}{q^{2k+1}}, $$ as $|q|\to\infty$, where $E_{2k+1}(0)$ are the special values of odd-order Euler polynomials at 0, and we also consider representations and bounds for the remainder of the above asymptotic expansion. In addition, we derive the asymptotic expansions for the higher order derivatives of $\zeta_{E}(s,q)$ with respect to its first argument $$\zeta_{E}^{(m)}(s,q)\equiv\frac{\partial^m}{\partial s^m}\zeta_E(s,q),$$ as $|q|\to\infty$. Finally, we also prove a new exact series representation of $\zeta_{E}(s,q)$.