Integral Representations for Multiple Ap\'ery-Like Series

TL;DR

This paper develops integral formulas for multiple Apéry-like series, connecting them with special functions and deriving new identities involving Dirichlet functions.

Contribution

It introduces novel integral representations for these series and establishes new identities linking them to Dirichlet eta, beta, and lambda functions.

Findings

Derived integral formulas involving polylogarithms and special functions.

Recovered known evaluations as special cases of the new formulas.

Established a new identity expressing series as combinations of Dirichlet eta values.

Abstract

We derive integral representations for six families of multiple Ap\'ery-like series using repeated integration by parts and Fourier expansions. The resulting formulas are expressed in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. As applications, we recover several known evaluations as special cases of our results, expressed in terms of Dirichlet eta, beta, and lambda functions. In addition, we obtain a new identity expressing a family of such series as linear combinations of products of Dirichlet eta values.