Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions

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 F-3(2) hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag-Leffler function and the Wright function.

Automated summary

This article calculates some infinite sums involving the digamma function and obtains interesting new results, such as parameter differentiation formulas for the beta incomplete function, reduction formulas for F-3(2) hypergeometric functions, and a definite integral not found in common literature. Additionally, the article applies these sums to calculate reduction formulas for the parameter differentiation of the Mittag-Leffler function and the Wright function.