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

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 $T=2l$ with integer $l\in {\mathbb{Z}}_{\ge 0}$ and obtain recurrence relations for $\beta (2k)$ and $\zeta (2k+1)$, where $\beta (z)$ is the Dirichlet beta function and $\zeta (z)$ 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.