Recurrence Relations for $β(2k)$ and $ζ(2k + 1)$

Abstract

In this work we study integrals of the form $\int_{0}^{\infty} \frac{\tanh(x)}{x} \sech(x)^{L} \exp(-Tx) dx, \quad T \in \mathbb{R}_{\geq 0}.$ We show that they can be represented as a sum of Hurwitz zeta derivatives with polynomial coefficients. As an application we evaluate these integrals for $T =2 l$ with integer $l \in \mathbb{Z}_{\geq 0}$ and obtain recurrence relations for $β(2k)$ and $ζ(2k + 1)$, where $β(z)$ is the Dirichlet beta function and $ζ(z)$ is the Riemann zeta function. For the negative $T$-derivative of the above integrals $\int_{0}^{\infty} \tanh(x)\sech(x)^{L} \exp(-Tx) dx$ we give simpler representations only involving digamma function values with polynomial coefficients.