New closed forms for a class of digamma series and integrals

TL;DR

This paper derives new closed-form expressions for a class of digamma series and integrals, revealing identities involving pi and providing tools for further mathematical exploration.

Contribution

It introduces novel closed forms for specific digamma series and integrals, expanding the understanding of such series and unveiling new identities involving pi.

Findings

Closed forms for digamma series involving \\psi functions.

New identities for generalized digamma series with pi terms.

Ten previously unstudied definite integrals over (0,1).

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.