On the Laplace-type transform and its applications

TL;DR

This paper introduces a new Laplace-type transform using the Gamma function, enabling the mapping of functions to functional sequences, and develops methods for solving difference equations and deriving combinatorial identities.

Contribution

It presents a novel integral transform that extends the Laplace transform's capabilities and connects it to difference operators and orthogonal polynomials.

Findings

New Laplace-type transform defined and characterized.

Method for solving difference equations using the transform.

Derivation of combinatorial identities via orthogonal polynomials.

Abstract

We use the Laplace transform and the Gamma function to introduce a new integral transform and name it the Laplace-type transform possessing the property of mapping a function to a functional sequence, which cannot be achieved by the Laplace transform. In addition, we construct a backward difference as a generalization of the backward difference operator $\nabla$, and connecting it to the Laplace-type transform, we deduce a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some elementary functions and their images is in the Appendix.