Recurrence Relations for $\beta(2k)$ and $\zeta(2k + 1)$

TL;DR

This paper derives explicit formulas for certain integrals involving hyperbolic functions in terms of derivatives of the Hurwitz zeta function, connecting them to special values of the Dirichlet beta and Riemann zeta functions, and provides recursive coefficient formulas.

Contribution

It introduces explicit expressions for integrals involving hyperbolic functions in terms of derivatives of the Hurwitz zeta function, linking them to special values of beta and zeta functions, with recursive coefficient formulas.

Findings

Explicit formulas for integrals in terms of Hurwitz zeta derivatives

Connections between integrals and special values of beta and zeta functions

Recursive formulas for coefficients in linear combinations

Abstract

In this work we study integrals of the form $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{L}\exp(-Tx)dx$. For $L \in \mathbb{N}$, $L \leq 4$ and $T \in \mathbb{R}$ we give explicit expressions in terms of derivatives of the Hurwitz zeta function at negative integers. We use these expressions to evaluate these integrals for $T \in \mathbb{N}_{0}$ exactly. For the special case $T = 0$ we give explicit evaluations for any $L \in \mathbb{N}$ based on the functional equations for $\beta(s)$ and $\zeta(s)$. As it turns out the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N + 1}dx$ is a linear combination of $\beta(2)$, ..., $\beta(2N + 2)$ and the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N}dx$ a linear combination of $\zeta(3)$, ..., $\zeta(2N + 1)$. We give recursive formulae for the coefficients in these linear combinations.