Identities for the Hankel transform and their applications

TL;DR

This paper explores identities involving the Hankel transform and related transforms, deriving new theorems that facilitate the evaluation of integrals of special functions and connecting these transforms with the Laplace transform.

Contribution

It introduces novel iteration identities for the Hankel and Widder transforms, leading to new Parseval-Goldstein type theorems and exchange identities involving the Laplace transform.

Findings

Derived iteration identities linking Hankel and Widder transforms

Established new Parseval-Goldstein type theorems for these transforms

Provided applications for evaluating integrals of special functions

Abstract

In the present paper the authors show that iterations of the Hankel transform with $\mathscr{K}_{\nu}$-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results presented here.