Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives (2103.15528v6)

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 $\zeta_{E}(s,q)$.