Fundamental relations between the Dirichlet beta function, euler numbers, and Riemann zeta function for positive integers

TL;DR

This paper introduces a novel definition of the Dirichlet beta function for positive integers, revealing new fundamental relations with Euler numbers, the Riemann zeta function, and the polygamma function, enhancing understanding of special constants.

Contribution

It presents the first redefinition of the Dirichlet beta function using the polygamma function, establishing new fundamental relations with key mathematical constants.

Findings

New definition of Dirichlet beta function for positive integers

Fundamental relations between polygamma, zeta, and Euler numbers

Method to derive special constants related to Dirichlet beta function

Abstract

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values, provides a general method for obtaining all special constants associated with Dirichlet beta function. We also show various new and fundamental relations between the polygamma function, Riemann zeta, the even-indexed euler numbers, the Dirichlet beta functions in a way never seen or imagined before.