An Alternative Approach to Computing $\beta(2k+1)$

TL;DR

This paper introduces a novel method for computing the odd integer values of the Dirichlet beta function using properties of Euler numbers and polynomials, providing new integral representations and building on previous Euler formula proofs.

Contribution

The paper develops a new approach to evaluate $eta(2k+1)$ and derive integral representations for $eta(2k)$, expanding computational techniques for special values of the Dirichlet beta function.

Findings

New method for $eta(2k+1)$ evaluation

Integral representation for $eta(2k)$

Extension of previous Euler formula proofs

Abstract

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $\beta(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses basic calculus and telescoping series. By a similar procedure, we also yield an integral representation of $\beta(2k)$. The idea of our proof adapts from a previous study by Ciaurri et al., where the authors introduced a new proof of Euler's formula for $\zeta(2k)$.