Integral representations, monotonic properties, and inequalities on extended central binomial coefficient

Abstract

In the paper, with the help of Kazarinoff's integral representation for the ratio of two gamma functions, with the aid the duplication formula of the digamma function, by virtue of integral representations of polygamma functions, and in the light of the L'Hôpital type monotonicity rule, the author describes extended binomial coefficients in terms of the gamma functions and the falling factorials, collects three integral representations of central binomial coefficients, establishes three integral representations of extended central binomial coefficients, proves decreasing and increasing properties of two power‐exponential functions involving extended (central) binomial coefficients, and derives several double and sharp inequalities for bounding extended (central) binomial coefficients, and compares these inequalities with known ones.

A Brief Survey and an Analytic Generalization of the Catalan Numbers and Their Integral Representations

“…The last three integral representations should provide effective tools for further studying central binomial coefficients ( 2n n ). These integral representations of central binomial coefficients ( 2n n ) were extended and investigated in Reference [47].…”