A class of index transforms generated by the Mellin and Laplace operators

TL;DR

This paper introduces a new class of index transforms derived from Mellin and Laplace operators, including novel transformations with hypergeometric kernels, and establishes their properties and inversion formulas.

Contribution

It constructs a new class of non-convolution index transforms based on Mellin and Laplace operators, with specific examples and inversion theorems.

Findings

Derived new index transforms with hypergeometric kernels

Established mapping properties and inversion formulas

Proved a new inversion theorem for the modified Kontorovich-Lebedev transform

Abstract

Classical integral representation of the Mellin type kernel in terms of the Laplace integral gives an idea to construct a new class of non-convolution (index) transforms. Particular examples give the Kontorovich-Lebedev-like transformation and new transformations with hypergeometric functions as kernels. Mapping properties and inversion formulas are obtained. Finally we prove a new inversion theorem for the modified Kontorovich-Lebedev transform