A note on a generalisation of a definite integral involving the Bessel function of the first kind

TL;DR

This paper generalizes a specific definite integral involving the Bessel function of the first kind, expressing it through the Fox-Wright function, and explores its implications and closed-form series evaluations.

Contribution

It introduces a new generalization of an integral involving Bessel functions and connects it to the Fox-Wright function, providing new closed-form series evaluations.

Findings

Integral expressed in terms of Fox-Wright function

Derived closed-form evaluations of infinite series

Explored consequences of the integral's representation

Abstract

We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_p\Psi_q(z)$ of one variable. Some consequences of this representation are explored by suitable choice of parameters. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced.