Parameterized summation relations for the Stieltjes constants

Abstract

The Stieltjes constants $\gamma_k(a)$ appear in the regular part of the Laurent expansion of the Hurwitz zeta function about its only polar singularity at $s=1$. We present multi-parameter summation relations for these constants that result from identities for the Hurwitz zeta function. We also present multi-parameter summation relations for functions $A_k(x)$ that may be expressed as sums over the Stieltjes constants. Integral representations, especially including Mellin transforms, play an important role. As a byproduct, reciprocity and other summatory relations for polygamma functions and Bernoulli polynomials may be obtained.