Hermite–Kampé de Fériet Hahn–Appell Polynomial Sequences: Characterisations, d-Orthogonality, and Zero Analysis

In this paper, we introduce and study Hermite–Kampé de Fériet Hahn–Appell polynomial sequences (HKdF Hahn–Appell sequences) within the unified framework of the Hahn difference operator. We establish three equivalent characterisations of these sequences: an operational definition via the Hahn difference condition, an explicit summation formula involving α-binomial coefficients and the falling Hahn basis, and a generating-function criterion. A three-term recurrence relation, a second-order difference equation, and the determinantal representation are also derived for these polynomials. The d-orthogonality of the family is characterised via truncation conditions on the recurrence coefficients {σk}; the generating functions of d-orthogonal members are identified as products of α-Pochhammer symbols, and the d-dimensional functional vector is expressed in closed form. As principal examples, the HKdF Hahn–Bernoulli, HKdF Hahn–Euler, and HKdF Hahn–Genocchi polynomial sequences are introduced, and for each family, four limiting cases are established. A unifying summary table is provided. Graphical and numerical analysis of the zero distribution for all three families confirms the analytical results and illustrates the convergence of zeros to the classical Hermite–Kampé de Fériet setting as the parameters approach their limiting values.