A double index transform with a product of Macdonald's functions revisited

Abstract

We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions $K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)$, where $(x, y) \in \mathbb{R}_+ \times \mathbb{R}_+$ and $i \tau, \tau \in \mathbb{R}_+$ is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function $K_{i \tau}(t)$ as a kernel.