Explicit Formulas For Generalized Polylogarithmic Integrals, Euler Sums, And BBP-Type Series

TL;DR

This paper derives explicit formulas for generalized polylogarithmic integrals, Euler sums, and BBP-type series using special functions, enhancing the understanding of their closed-form representations.

Contribution

It provides new explicit formulas for polylogarithmic integrals and related series in terms of the Lerch transcendent and other special functions.

Findings

Closed-form expressions for polylogarithmic integrals in terms of Lerch transcendent

Connections between integrals, Euler sums, and BBP-type series

Reduction to well-known special functions like zeta and polygamma

Abstract

This paper explores closed-form expressions for some polylogarithm integrals with integrands containing five parameters. These closed form expressions are given in terms of the Lerch transcendent function, which reduces, in some cases, to well-known special functions, such as the Riemann zeta, Dirichlet eta, Dirichlet beta, the Hurwitz zeta, and the polygamma functions. Related Euler sums and BBP-type series will also be discussed.