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

TL;DR

The paper addresses representing Catalan's constant $G=beta(2)$ and Apéry's constant $xi(3)$ using the half hyperbolic secant distribution, and relates these constants to Euler numbers $E_{2k}$ and the Dirichlet beta function. It develops integral (simple and double) and series representations through moments, convolutions, and mixtures, while linking these constructs to inequality measures such as the Lorenz curve, Gini, and Theil, as well as Basel-type sums. Key contributions include new moment-based representations of $G$, fresh integral forms for Euler numbers, multiple Apéry representations, and several mixture- and inequality-based identities that yield new bounds and computational tools. The results provide a versatile bridge between analytic number theory and probabilistic/income-inequality frameworks, with potential implications for numerical evaluation and cross-disciplinary applications.

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.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9