Hyperharmonic zeta and eta functions via contour integral

In this paper, we investigate the hyperharmonic zeta and eta functions via the Hankel contour integral. Employing contour integral representations, we find the values at the negative even integers of the hyperharmonic eta function and describe Laurent expansions at simple poles of the hyperharmonic zeta function. Moreover, for the Stieltjes constants \({\gamma }_{{h}^{(r)}}\) ${\gamma }_{h^{(r)}}$ (m), we give an alternative closed-form expression in terms of some special constants and certain integrals. In addition, we show that the values at the positive integers of the hyperharmonic eta function can be expressed in closed-form evaluations in terms of zeta values and log-sine integrals.