On $(p,q)$-Appell Polynomials

TL;DR

This paper introduces and characterizes $(p,q)$-Appell polynomials, exploring their algebraic properties, recurrence relations, and specific examples including $(p,q)$-Hermite polynomials with their difference equations.

Contribution

The paper provides a new framework for $(p,q)$-Appell polynomials, including their characterizations, algebraic structure, and explicit examples with recurrence and difference equations.

Findings

Defined $(p,q)$-Appell polynomial sets.

Derived recurrence relations and difference equations.

Presented examples including $(p,q)$-Hermite polynomials.

Abstract

We introduce polynomial sets of $(p,q)$-Appell type and give some of their characterizations. The algebraic properties of the set of all polynomial sequences of $(p,q)$-Appell type are studied. Next, we give a recurrence relation and a $(p,q)$-difference equation for those polynomials. Finally, some examples of polynomial sequences of $(p,q)$-Appell type are given, particularly, a set of $(p,q)$-Hermite polynomials is given and their three-term recurrence relation and a second order homogeneous $(p,q)$-difference equation are provided.