Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

TL;DR

This paper investigates the hyperharmonic zeta function's Laurent expansion, introduces new constants called hyperharmonic Stieltjes constants, and expresses these constants and related integrals in terms of special constants.

Contribution

It extends the theory of Stieltjes constants to the hyperharmonic zeta function and relates these constants to integrals involving generalized exponential functions.

Findings

Defined hyperharmonic Stieltjes constants in Laurent expansion

Expressed these constants as finite combinations of special constants

Connected constants to integrals involving generalized exponential functions

Abstract

In this paper, we consider meromorphic extension of the function \[ \zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_{n}^{(r)}$ are the hyperharmonic numbers. We establish certain constants, denoted $\gamma_{h^{\left( r\right) }}\left( m\right) $, which naturally occur in the Laurent expansion of $\zeta_{h^{\left( r\right) }}\left( s\right) $. Moreover, we show that the constants $\gamma_{h^{\left( r\right) }}\left( m\right) $ and integrals involving generalized exponential integral can be written as a finite combination of some special constants.