Umbral Calculus, a Different Mathematical Language

TL;DR

This thesis explores umbral calculus as a symbolic mathematical language that simplifies the derivation and understanding of special functions and polynomials through operational methods and formal symbolic techniques.

Contribution

It introduces a formalism of umbral calculus that offers a new symbolic framework for translating and simplifying the theory of special functions and polynomials.

Findings

Simplifies derivation of properties of special functions

Provides rules for replacing transcendental functions with elementary functions

Enhances understanding of operational methods in mathematical analysis

Abstract

This thesis is intended to provide an account of the theory and applications of Operational Methods that allow the "translation" of the theory of special functions and polynomials into a "different" mathematical language. The language we are referring to is that of symbolic methods, largely based on a formalism of umbral type which provides a tremendous simplification of the derivation of the associated properties. The strategy we will follow is that of establishing the rules to replace higher trascendental functions in terms of elementary functions and to take advantage from such a recasting.