On the product formula and convolution associated with the index Whittaker transform

TL;DR

This paper develops a new product formula and convolution operator related to the index Whittaker transform, generalizing existing convolutions and establishing properties like Banach algebra structure and Wiener-Lévy type theorems.

Contribution

It introduces a novel product formula for the Whittaker function and defines a positivity-preserving convolution operator, extending the Kontorovich-Lebedev convolution.

Findings

Established a product formula independent of the second parameter.

Defined a positivity-preserving convolution operator for the index Whittaker transform.

Proved a Wiener-Lévy type theorem and existence of solutions for convolution integral equations.

Abstract

We deduce a product formula for the Whittaker $W$ function whose kernel does not depend on the second parameter. Making use of this formula, we define the positivity-preserving convolution operator associated with the index Whittaker transform, which is seen to be a direct generalization of the Kontorovich-Lebedev convolution. The mapping properties of this convolution operator are investigated; in particular, a Banach algebra property is established and then applied to yield an analogue of the Wiener-L\'evy theorem for the index Whittaker transform. We show how our results can be used to prove the existence of a unique solution for a class of convolution-type integral equations.