Algebraic Characterizations and Properties of Bivariate 2D q‐Hermite–Based Appell Sequences

ABSTRACT This paper presents a novel family of bivariate two‐dimensional ( ) ‐Hermite–based Appell polynomials, introducing their construction along with explicit examples to illustrate their structure. The study explores key mathematical properties of these polynomials, including explicit forms and determinant‐based representations that provide a foundational understanding of their algebraic framework. Central to the discussion is the establishment of the monomiality principle, which forms a crucial basis for deriving several other properties. The paper also examines ‐recurrence relations and ‐difference equations, further highlighting the interplay between these properties and the underlying ‐Hermite–based framework. In addition, the findings are extended to special classes of bivariate polynomials, namely the ‐Hermite–based Bernoulli, Euler, and Genocchi polynomials, demonstrating the broader applicability and versatility of the proposed approach. These results enrich the existing theory of ‐polynomials and provide new tools for analyzing and generalizing classical polynomial families in the context of ‐calculus.