Sums involving the digamma function connected to the incomplete beta function and the Bessel functions

TL;DR

This paper derives closed-form expressions for infinite sums involving the digamma function, linking them to the incomplete beta and Bessel functions, and applies these results to special functions like Mittag-Leffler and Wright functions.

Contribution

It introduces new closed-form sums involving the digamma function and connects them to special functions, providing new formulas and integral representations.

Findings

New parameter differentiation formulas for the incomplete beta function

Reduction formulas for hypergeometric functions $_{3}F_{2}$

A novel definite integral not previously tabulated

Abstract

We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of $_{3}F_{2}$ hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of some sums involving the digamma function, we have calculated some redution formulas for the parameter differentiation of the Mittag-Leffler function and the Wright function.