A New Formula for The Values of Dirichlet Beta Function at Odd Positive Integers Based on The WZ Method

TL;DR

This paper introduces a novel formula for calculating the Dirichlet beta function at odd positive integers using the Wilf-Zeilberger (WZ) method, advancing the analytical tools available for special function evaluation.

Contribution

It presents a new formula for the Dirichlet beta function at odd integers derived via the WZ method, which is a novel approach in this context.

Findings

Derived a new formula for β(s) at odd positive integers.

Utilized the WZ method to establish the formula.

Provides a potentially more efficient way to compute β(s).

Abstract

By using the related results in the WZ theory, a new (as far as I know) formula for the values of Dirichlet beta function $\beta (s) = \sum\limits_{n = 1}^{+ \infty} {\frac{(-1)^{n - 1}}{(2n - 1)^s}} $ (where $Re(s) > 0$) at odd positive integers was given.