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

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 $k$.