Polynomials with r-Lah coefficient and hyperharmonic numbers

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers

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

In this paper, we consider meromorphic extension of the function $$\zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r,$$ (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