Some evaluations of infinite series involving Dirichlet type parametric harmonic numbers (Q6589629)

In this paper under review, the authors extend the parametric digamma functions \(\Psi(-s, a)\) and \(\Psi(-s; A)\) introduced by \textit{C. Xu} in [Int. J. Number Theory 15, No. 7, 1531--1546 (2019; Zbl 1459.41016); Kyushu J. Math. 75, No. 2, 295-322 (2021; Zbl 1492.65005)] to the general function \(\Psi(-s; A, a)\) defined by \N\[\N\Psi(-s; A, a)+\gamma=\frac{a_0}{s-a}+\sum_{k=1}^\infty \left(\frac{a_k}{k+a}-\frac{a_k}{k+a-s}\right), \quad (-1<a<1). \N\]\NThe function \(\Psi(-s; A, a)\) is meromorphic in the entire complex plane with a simple pole at \(z = n + a\) for each nonnegative integer \(n\). The authors use the method of contour integrations involving \(\Psi(-s; A, a)\) and residue theorem to establish some explicit relations of infinite series involving Dirichlet type parametric harmonic numbers which are of the form \N\[\N\sum_{k=1}^\infty \frac{a_k}{(k+a)^p}, \quad p,\, n\in \mathbb{N}, \, a\in \mathbb{C}\backslash \mathbb{N}^- .\N\]\NThen by applying these obtained formulas, they establish some explicit relations of parametric linear Euler sums and some special functions such as trigonometric functions, digamma functions and Hurwitz zeta functions.