A Unified Operational and Algebraic Approach of $\Delta_h$-Hybrid Polynomials Associated with Appell Sequences

Abstract

This study introduces a new class of $\Delta_h$ Legendre-Laguerre-Appell polynomials, constructed through the synergy of the monomiality framework and operational calculus. A comprehensive exploration is carried out, beginning with the formulation of their generating function, followed by the derivation of explicit representations and recurrence schemes. Notably, a determinantal structure for these polynomials is also established and illustrated through representative examples. The work further investigates how this polynomial family interrelates with well-known $\Delta_h$-variants of classical polynomials, including the Bernoulli, Euler, and Genocchi types. Through these connections and properties, the results not only deepen our understanding of the algebraic and analytic behaviour of the $\Delta_h$ Legendre-Laguerre-Appell polynomials but also highlight their potential applications in broader areas of discrete mathematics and operational theory.