Hypergeometric summation representations of the Stieltjes constants

TL;DR

This paper presents new hypergeometric summation formulas for the Stieltjes constants, generalizing known representations of the Euler constant, and discusses their convergence and asymptotic properties.

Contribution

It introduces novel summation representations of the Stieltjes constants using hypergeometric functions, expanding the analytical tools available for their study.

Findings

Derived summation formulas for Stieltjes constants using hypergeometric functions

Generalized the representation of Euler constant to Stieltjes constants

Provided asymptotic analysis and evaluation of related integrals

Abstract

The Stieltjes constants $\gamma_k$ appear in the regular part of the Laurent expansion of the Riemman and Hurwitz zeta functions. We demonstrate that these coefficients may be written as certain summations over mathematical constants and specialized hypergeometric functions $_pF_{p+1}$. This family of results generalizes a representation of the Euler constant in terms of a summation over values of the trigonometric integrals Si or Ci. The series representations are suitable for acceleration. As byproducts, we evaluate certain sine-logarithm integrals and present the leading asymptotic form of the particular $_pF_{p+1}$ functions.