2‐Variable <i>q</i> ‐Legendre‐Based Appell Polynomials and Their Applications

ABSTRACT In this paper, we introduce the theory of two‐variable ‐Legendre‐based Appell polynomials through the framework of the zeroth‐order ‐Tricomi functions. These polynomials are investigated via their generating functions, series expansions, and determinant representations. Furthermore, employing the principles of ‐quasi‐monomiality, we establish that these polynomials possess the ‐quasi‐monomial property, deriving several operational representations and formulating the corresponding ‐differential equations. Several illustrative examples are presented to elucidate the theory of ‐Legendre‐based Appell polynomials and to highlight the properties established above. Finally, graphical interpretations are provided, including plots, surface visualizations, and depictions of the spatial distribution of scattered zeros in the ‐plane for selected two‐variable ‐Legendre‐based Appell polynomials.