Operational vs. Umbral Methods and Borel Transform

TL;DR

This paper explores the use of differintegral methods and Borel transforms to connect umbral and operational techniques, providing new tools for evaluating integrals and generating functions of special functions.

Contribution

It introduces a novel approach combining umbral and operational methods via Borel transforms for analyzing special functions.

Findings

Effective link between umbral and operational methods established

New integral evaluation techniques for special functions developed

Enhanced summation methods for generating functions introduced

Abstract

Differintegral methods, currently exploited in calculus, provide a fairly unexhausted source of tools to be applied to a wide class of problems involving the theory of special functions and not only. The use of integral transforms of Borel type and the associated formalism will be shown to be an effective means, allowing a link between umbral and operational methods. We merge these two points of view to get a new and efficient method to obtain integrals of special functions and the summation of the associated generating functions as well.