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 $\mathcal{A}\left(t\right){(1-\eta t^r)}^{-1}{e}_q^{wt}\left(\zeta t\right)$, where $\mathcal{A}\left(t\right)$ is a normalized determining function and $e_q^{wt}$ 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 $\mathcal{A}\left(t\right)=1$, numerical computations for $q=1/2$, $r=2$, $\eta =1$, 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.