New closed forms for a class of digamma series and integrals

Abstract

The pursuit of closed forms for infinite series has long been a focal point of research. In this paper, we contribute to this endeavor by presenting closed forms for the class of digamma series: \[\sum_{k=1}^\infty \frac{\psi\left(\frac{2k+2n+5}{4}\right) - \psi\left(\frac{2k+2n+3}{4}\right)}{(2k + \alpha)^2},\] \[\sum_{k=1}^\infty (-1)^k \frac{\psi\left(\frac{2k+2n+5}{4}\right) - \psi\left(\frac{2k+2n+3}{4}\right)}{(2k + \alpha)^2},\] for all non-negative integers $\alpha$ and $n$. In addition to providing closed forms for these series, we unveil new identities for various generalized digamma series in the elegant form $a_0 + a_1 \pi + a_2 \pi^2 + a_3 \pi^3$, where $a_0, \ldots, a_3$ are real-valued constants determined by our formulas. Furthermore, we present ten definite integrals over the interval $(0, 1)$ that have not been previously studied in the literature and appear to be nearly impossible to evaluate. Combining these series and integrals can lead to the discovery of even more new results. Our findings contribute to the study of closed forms for infinite series and integrals, offering novel results and potential avenues for further exploration.