A New Generalization of Two-Variable q-Gould–Hopper–Hahn–Appell Polynomials via Quantum q-Calculus

In this paper, we introduce a class of two-variable q-Gould–Hopper–Hahn–Appell polynomials obtained by combining a Hahn-type q-Gould–Hopper kernel with a general q-Appell factor. Using a formal generating function approach, we derive explicit series representations, connection and convolution formulas, higher and mixed q-derivative identities, inverse operational formulas, two-direction shift relations, and a determinant representation in the non-singular case A0,q≠0. We then develop an operational description of the family and establish its quasi-monomial structure, which yields the corresponding lowering operator, an induced raising operator, and the associated q-differential equation. We also discuss reductions to previously known subclasses, distinguish clearly between non-singular and singular q-Appell specializations, and include representative examples illustrating how an Appell factor affects the low-degree zero pattern.