In the present work, we introduce and study a new two-parameter generalization of Legendre-based Appell polynomials, defined through an explicit representation that unifies classical Legendre structures with the Appell polynomial framework. Starting from a generating function, we derive a three-term recurrence relation, a degree-lowering operator, an integro-partial degree-raising operator, and a corresponding integro-partial differential equation satisfied by the new family. A determinant representation is established via Cramer’s rule applied to the Cauchy-product expansion of the generating function. Several subfamilies of independent interest arise naturally as special cases, namely, Legendre-based Hermite–Frobenius–Euler polynomials, Legendre-based Miller–Lee polynomials, and both the probabilist’s and physicist’s variants of Legendre-based bi-variate Hermite polynomials. For each subfamily we record the corresponding recurrence relations, shift operators, differential equations, and determinant forms, and we illustrate the behavior of selected members through three-dimensional surface plots and real-root distribution diagrams. The framework presented here extends several constructions available in the recent literature and points to natural directions for future work, including connections with q-series, combinatorial identities, and symbolic-computation methods, which are outlined in the concluding section.
Alhamzi et al. (Fri,) studied this question.