New representations of Catalan’s constant obtained from the half hyperbolic secant distribution

Abstract

Derived from the half hyperbolic secant distribution, new expressions and bounds for Catalan’s constant are presented. They are obtained by using expressions for the Lorenz curve, the Gini and Theil indices, convolutions, and a mixture of distributions, among other approaches. The new expressions are presented both in terms of integral (simple and double) representation and also as an interesting series representation. In addition, some integral representations of Apéry’s constant and Euler numbers are obtained.