Generalized Transforms and Special Functions

TL;DR

This paper explores the properties of various transforms, especially the binomial transform and its generalizations, using operational methods and Fourier techniques to analyze their relationship with special functions.

Contribution

It introduces a general Fourier-based method for studying functions of operators and their connection to special functions, extending the understanding of transforms.

Findings

Characterization of binomial and generalized transforms

Development of a Fourier-based operational method

Insights into the relationship between transforms and special functions

Abstract

We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A general method, based on the use of the Fourier transform technique, is proposed for the study of the properties of functions of operators.