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 $k$.

10 pages, no figures

References in corpus (3)

• Expansions of generalized Euler's constants into the series of polynomials in ${\pi }^{-2}$ and into the formal enveloping series with rational coefficients only
• New results on the Stieltjes constants: Asymptotic and exact evaluation
• Functional equations for the Stieltjes constants