Contour Integration and Cyclotomic Ap\'ery-Like Series Involving Generalized Binomial Coefficients

TL;DR

This paper introduces a contour integration method to analyze cyclotomic Apéry-like series involving generalized binomial coefficients, expressing them in terms of polylogarithms and zeta values, and deriving new identities and results.

Contribution

It presents a novel contour integration approach to express cyclotomic Apéry-like series in terms of special functions, recovering known results and establishing new identities.

Findings

Expressed series in terms of multiple polylogarithms and zeta values

Derived identities for multiple polylogarithm functions

Provided examples and corollaries illustrating the method

Abstract

In this paper, we present a method based on contour integration to investigate a class of cyclotomic parametric Ap\'ery-like series. The general term of such series involves a parametric central binomial coefficient, which is defined via the Gamma function. Using this approach, we express a family of cyclotomic Ap\'ery-like series in terms of multiple polylogarithms, cyclotomic Hurwitz zeta values, Riemann zeta values and $\log(2)$. In particular, we provide several illustrative examples and corollaries, which enable us to recover a number of known results on Ap\'ery-like series. At the same time, we have also left open two questions regarding Ap\'ery-like series. Moreover, by considering integrals of the generating function for Fuss-Catalan numbers, we derive an alternative expression for a classical Ap\'ery-like series. Combining this with known results allows us to establish several identities for multiple polylogarithm functions.