Deformed Bivariate $q$-Appell Polynomials
Ronald Orozco López
TL;DR
This work develops a unified framework for deformed bivariate $q$-Appell polynomials of order $ abla$, defined via $( abla_q(t))^{ abla} e_q(tx)e_q(ty,u)$, connecting them to deformed homogeneous polynomials $ ext{R}_n$ and deriving Mehler's and Rogers-type formulas for quasi-$q$-Appell families. It establishes multiple equivalent characterizations, including generating-function forms, explicit closed-form expansions, and operator/differential relations, while also presenting an algebraic structure that forms a commutative group $ rak A(q;u)$ under a $*$-product. The paper further constructs deformed Appell operators and shows how quasi- and trivariate polynomials arise from these operators, with direct links back to the original deformed polynomials. Finally, it demonstrates concrete instances by defining $u$-deformed $q$-Bernoulli, $q$-Euler, and $q$-Genocchi polynomials and numbers, including their bivariate extensions, highlighting the versatility and potential applications in combinatorics and special functions.
Abstract
In this paper, we introduce bivariate polynomial sets of deformed $q$-Appell type, and we study the algebraic properties of these sets. We show the relation between deformed bivariate $q$-Appell polynomials and deformed homogeneous polynomials. Next, we give some of their characterizations and algebraic structure. Then, we introduce the deformed $q$-Appell operators and obtain Mehler's and Rogers-type formulas of quasi-$q$-Appell polynomials. Finally, some examples of polynomial sequences of deformed $q$-Appell type are given: Bernoulli, Euler, and Genocchi types.