A $q$-Umbral Approach to $q$-Appell Polynomials

TL;DR

This paper explores the properties of $q$-Appell polynomials using $q$-Umbral calculus, focusing on $q$-Genocchi numbers and polynomials, and demonstrates their linear combination representation.

Contribution

It introduces a $q$-Umbral calculus framework to analyze $q$-Appell polynomials and shows how arbitrary polynomials can be expressed via $q$-Genocchi polynomials.

Findings

Any polynomial can be written as a linear combination of $q$-Genocchi polynomials.

Similar properties are found for other $q$-Appell polynomials.

The $q$-Umbral approach provides new insights into the structure of these polynomials.

Abstract

In this paper we aim to specify some characteristics of the so called family of $q$-Appell Polynomials by using $q$-Umbral calculus. Next in our study, we focus on $q$-Genocchi numbers and polynomials as a famous member of this family. To do this, firstly we show that any arbitrary polynomial can be written based on a linear combination of $q$-Genocchi polynomials. Finally, we approach to the point that similar properties can be found for the other members of the class of $q$-Appell polynomials.