New closed forms for a dilogarithmic integral, related integrals, and series

Abstract

In this study, we present a new closed form for the generalized integral $${\int }_0^1\frac{{\mathrm{Li}}_2(z)\ln (1+az)}{z}\mathrm{d}z,$$ where $a\in \mathbb{C}\setminus (-\infty ,-1)$ and ${\mathrm{Li}}_2(z)$ is the dilogarithm function. This generalization is achieved by leveraging our established findings in conjunction with Vălean's results. Furthermore, we provide explicit closed forms for associated integrals, prove a transformation formula for double infinite series, expressing them as the sum of the square of an infinite series and another infinite series. We utilize this relationship to derive a novel closed form for the generalized series $$\sum _{k=1}^{\infty }\frac{\hat{I}\P \left(m,\frac{rk-s}{r}\right)}{(rk-s{)}^m},$$ for $\Re (m)>1$, $r,s\in \mathbb{C}$, where $r\ne 0$, $rk\ne s$, for any positive integer $k$, and $\hat{I}\P (s,z)$ denotes the Hurwitz zeta function. Utilizing Hermite's integral representation for $\hat{I}\P (s,z)$, we derive a family of integrals from this series.

25 pages