Further Classes of Series Involving Central Binomial Coefficients

TL;DR

This paper extends known identities involving series with central binomial coefficients to complex parameters, using advanced mathematical tools like Bell polynomials and gamma functions, thereby broadening the understanding of such series.

Contribution

It introduces new generalizations of series involving central binomial coefficients for all complex parameters, employing diverse methods and special functions.

Findings

Extended identities to complex parameters

Developed new integral representations

Utilized gamma and polygamma functions extensively

Abstract

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including exponential Bell polynomials and integral representations, to further extend these results. Throughout the paper, we make extensive use of the gamma and polygamma functions and their properties.