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

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