Note on the Stieltjes constants: series with Stirling numbers of the first kind

TL;DR

This paper generalizes series representations for the Stieltjes constants using Stirling numbers, providing new integral and series formulas and highlighting recent asymptotic developments for better accuracy.

Contribution

It extends existing series and integral formulas for Stieltjes constants to the general case of a parameter a, incorporating Stirling numbers of the first kind.

Findings

Derived new series and integral representations for γ_k(a)

Connected recent asymptotic formulas to practical computation

Enhanced understanding of Stieltjes constants' behavior

Abstract

The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We generalize the integral and Stirling number series results of [4] for $\gamma_k(a=1)$. Along the way, we point out another recent asymptotic development for $\gamma_k(a)$ which provides convenient and accurate results for even modest values of $k$.