Double series expression for the Stieltjes constants

TL;DR

This paper derives a double infinite series expression for the Stieltjes constants, linking them to the Hurwitz zeta function and providing a new series representation for these constants and the zeta function.

Contribution

It introduces a novel double series representation for the Stieltjes constants and the Hurwitz zeta function, expanding the analytical tools available for these special functions.

Findings

Derived a double series for Stieltjes constants

Connected the case $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{γ}}}_k(1)= ext{γ}}}$ to Brun's series for the Riemann zeta

Provided a parameterized double series for the Hurwitz zeta function

Abstract

We present expressions in terms of a double infinite series for the Stieltjes constants $\gamma_k(a)$. These constants appear in the regular part of the Laurent expansion for the Hurwitz zeta function. We show that the case $\gamma_k(1)=\gamma$ corresponds to a series representation for the Riemann zeta function given much earlier by Brun. As a byproduct, we obtain a parameterized double series representation of the Hurwitz zeta function.