Can Umbral and $q$-calculus be merged?

TL;DR

This paper reformulates $q$-calculus using umbral calculus, revealing new insights and applying the approach to $q$-special functions, integrals, and polynomials, thus bridging two mathematical frameworks.

Contribution

It introduces a novel reformulation of $q$-calculus via umbral calculus, enabling new derivations and understanding of $q$-special functions and related integrals.

Findings

New formulation of $q$-calculus using umbral calculus

Derivation of $q$-special functions and integrals with the new approach

Enhanced understanding of $q$-polynomials and their properties

Abstract

The $q$-calculus is reformulated in terms of the umbral calculus and of the associated operational formalism. We show that new and interesting elements emerge from such a restyling. The proposed technique is applied to a different formulations of $q$ special functions, to the derivation of integrals involving ordinary and $q$-functions and to the study of $q$-special functions and polynomials.