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.

15 pages, no figures, 2 new Propositions, Corollaries, and references added

Cited by in corpus (2)

• Summatory relations and prime products for the Stieltjes constants, and other related results
• Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall