  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.
