On a Weighted Series of the Hurwitz Zeta Function

In this note we prove that for all $a\in \mathbb{N}$, $x\in {\mathbb{R}}_{+}\cup \{0\}$, and $s\in \mathbb{C}$ with $\mathrm{\Re }(s)>a+2$, the (alternating) weighted series of the Hurwitz zeta function, $$\sum _{k\ge 1}(\pm 1{)}^k(k+x{)}^a\zeta (s,k+x),$$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zenaïdi’s theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.