Operational versus umbral methods and the Borel transform

TL;DR

This paper explores the use of Borel transform-based integral methods to connect umbral and operational techniques, leading to new analytical tools for special functions and their generating functions.

Contribution

It introduces a novel approach that merges umbral and operational methods via Borel transforms to derive integrals and sum generating functions more effectively.

Findings

Develops new integral formulas for special functions

Provides methods for summing generating functions

Establishes a bridge between umbral and operational techniques

Abstract

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.