Classical Apéry-Like Series and Their Cyclotomic Parametric Analogues via Contour Integration
Ce Xu
TL;DR
The paper develops a contour-integral framework to analyze cyclotomic parametric Apéry-like series, whose terms involve a parametric central binomial coefficient defined via the Gamma function. It derives explicit identities expressing these series in terms of $\log(2)$, ordinary and cyclotomic zeta values, and multiple polylogarithms, with residue calculus guided by Bell polynomials and an extended trigonometric function $\Phi$. A second related Apéry-like series is treated similarly, yielding corollaries that recover known results and connect to Fuss–Catalan generating functions. Additionally, the authors present an alternative expression for a class of Euler-Apéry-like series derived from Fuss–Catalan integrals, enabling further identities among multiple polylogarithms and suggesting avenues for generalization and open questions.
Abstract
In this paper, we present a method based on contour integration to investigate a class of cyclotomic parametric Apéry-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 Apéry-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 Apéry-like series. At the same time, we have also left open two questions regarding Apéry-like series. Moreover, by considering integrals of the generating function for Fuss-Catalan numbers, we derive an alternative expression for a classical Apéry-like series. Combining this with known results allows us to establish several identities for multiple polylogarithm functions.