Evaluation of q-Truncated-Exponential-Based Hahn–Appell Polynomials in the Framework of Quantum Calculus

This paper develops a two-variable truncated-exponential-based Hahn–Appell polynomial family of order r in quantum q-calculus, generated by A(t)(1−ηtr)−1eqwt(ζt), where A(t) is a normalized determining function and eqwt denotes the Hahn-type q-exponential. From this relation, we derive connections with the underlying Hahn–Appell basis, the finite expansion induced by the truncation factor, the Hahn lowering property, operational resolvents, step-r recurrences, determinant formulas, and Hahn-factorial representations. We also obtain origin values, normalized monicity, parameter-connection and parameter-differentiation identities, quasi-monomial operators, and the associated Hahn-grid difference equation. Limiting and special cases recover known q-truncated-exponential and classical truncated-exponential Appell constructions. For the unit-Appell specialization A(t)=1, numerical computations for q=1/2, r=2, η=1, and w=1 provide explicit polynomials, zero tables, diagnostics, and plots that illustrate parameter sensitivity, Hahn-grid deformation, conjugate symmetry of non-real zeros, origin zeros in odd degrees, and the observed layered rightward spread of zero clouds.