A series of definite integrals involving upper incomplete Gamma functions

TL;DR

This paper derives a new summation formula for derivatives of Bessel and Struve functions using probability theory and upper incomplete Gamma functions, revealing novel connections between special functions and probability distributions.

Contribution

It introduces a novel probabilistic approach to evaluate series of integrals involving upper incomplete Gamma functions and derives new formulas for derivatives of special functions.

Findings

New summation formula for derivatives of Bessel and Struve functions.

Probabilistic method linking incomplete Gamma functions with special functions.

Application of Beta mixture distributions in integral evaluations.

Abstract

Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial role. We also give an interesting application of our result, namely, a new summation formula for some derivatives of the Bessel functions of the first kind and the Struve functions with respect to the order.