Series involving parametric harmonic zeta function

Abstract

This paper investigates the analytic structure of the parametric harmonic zeta function \[ ζ_{H}\left( s,a,b\right) =\sum_{n=0}^{\infty}\frac{H_{n}\left( a\right) }{\left( n+b\right) ^{s}}, \] where $H_{n}\left( a\right) $ denotes the $n$th generalized harmonic number. We first establish the meromorphic continuation of $ζ_{H}\left(s,a,b\right) $ to the whole complex plane, except for a set of poles, and explicitly determine the residues at its poles. Secondly, we derive the Taylor expansion of $ζ_{H}\left( s,a,b+t\right) $ around $t=0$, serving as a generating function that enables generalizations of several classical identities of Landau, Singh-Verma, and Srivastava to the harmonic zeta setting. We then develop explicit expressions for the associated harmonic Stieltjes constants $γ_{H,-v}\left( m,a,b\right) ,$ $v\in\mathbb{N} \cup\left\{ -1,0\right\} $. These formulas include cases for which no closed forms were previously available, such as $γ_{H,-v}\left( m,a\right) $ and $γ_{H,-v}\left( m\right) ,$ $v\in\mathbb{N}\cup\left\{ 0\right\} $. Finally, we introduce a new special function, the harmonic digamma function, and show that it shares key analytic properties with the classical digamma function, including difference equations, derivative identities, and Taylor series expansion.