New closed forms for a dilogarithmic integral, relatedintegrals, and series

Resumen

In this study, we present a new closed form for the generalized integral
\[
\int_0^1 \frac{\text{Li}_2(z) \ln(1 + a z)}{z} \, dz,
\]
where \( a \in \mathbb{C} \setminus (-\infty, -1) \) and \(\text{Li}_2(z)\) 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
\[
\sum_{k=1}^\infty \frac{\zeta\left(m, \frac{rk - s}{r}\right)}{(rk - s)^m},
\]
for \(\Re(m) > 1\), \(r, s \in \mathbb{C}\), where \(r \neq 0\), \(rk \neq s\), for any positive integer \(k\), and \(\zeta(s, z)\) denotes the Hurwitz zeta function. Utilizing Hermite’s integral representation for \(\zeta(s, z)\), we derive a family of integrals from this series.