Addison-type series representation for the Stieltjes constants

TL;DR

This paper extends Addison's technique to derive series representations for the Stieltjes constants, which are coefficients in the Laurent expansion of the Hurwitz zeta function, providing new ways to express these mathematical constants.

Contribution

The paper introduces a generalized method based on Addison's technique to obtain series representations for the Stieltjes constants, expanding previous approaches for the Euler constant.

Findings

Derived new series representations for Stieltjes constants

Generalized Addison's technique to a broader class of constants

Provided alternative expressions for the Euler constant

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.