Addison-type series representation for the Stieltjes constants

Abstract

The Stieltjes constants $\gamma_k(a)$ appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about its only pole at $s=1$. We generalize a technique of Addison for the Euler constant $\gamma=\gamma_0(1)$ to show its application to finding series representations for these constants. Other generalizations of representations of $\gamma$ are given.