New Representations of Catalan's Constant, Apery's Constant and the Euler Numbers Obtained from the Half Hyperbolic Secant Distribution

TL;DR

This paper introduces novel integral and series representations for Catalan's constant, Apery's constant, and Euler numbers using the half hyperbolic secant distribution, expanding mathematical understanding of these constants.

Contribution

It provides new expressions and bounds for these constants derived from the half hyperbolic secant distribution, including integral and series forms, some of which are novel.

Findings

New integral and series representations for Catalan's and Apery's constants.

Derived bounds for the constants using distribution-based approaches.

New integral representations for Euler's numbers.

Abstract

New expressions and bounds for Catalan's and Apery's constants, derived from the half hyperbolic secant distribution, are presented. These constants 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. Some of these features are well known, while others are new. In addition, some integral representations of Euler's numbers are obtained.