Further Classes of Series Involving Central Binomial Coefficients

TL;DR

This work extends classical series involving central binomial coefficients to general complex parameters via gamma, beta, and polygamma techniques, and develops a comprehensive framework that unifies and broadens prior results. The core contributions include a gamma-quotient closed form for the base sum, a detailed treatment of singular values, and explicit evaluations for sums with squared and cubed linear denominators. The introduction of exponential Bell polynomials enables arbitrary positive powers in the denominator, with results connected to harmonic numbers, zeta values, and Hurwitz zeta functions, complemented by integral representations and generating-function identities. Collectively, the methods provide a versatile toolkit for deriving new closed forms and integral representations in analytic combinatorics and special-function theory, with broad potential for further extensions and applications.

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.

Theorems & Definitions (34)

• Theorem 1.1
• proof : Proof of Theorem \ref{['thm:1.1']}
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:3.1']}