On the Stieltjes constants with respect to harmonic zeta functions

The aim of this paper is to investigate harmonic Stieltjes constants occurring in the Laurent expansions of the function $${\zeta }_H(s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(n+a)}^s}\sum _{k=0}^n\frac{1}{k+a},\mathrm{Re}\left(s\right)>1,$$ which we call harmonic Hurwitz zeta function. In particular evaluation formulas for the harmonic Stieltjes constants ${\gamma }_H(m,1\mathrm{/}2)$ and ${\gamma }_H(m,1)$ are presented.