Families of differential equations and determinant forms of the generalized Legendre-Appell and related polynomials

In this study, we propose an extended form of the Legendre-based Appell polynomial families and examine their essential analytical properties. By employing the quasi-monomial approach, we establish the corresponding recurrence relations, multiplicative and derivative operators, together with the governing differential equations. Moreover, we formulate both the series and determinant representations for this newly constructed class of polynomials. Within this framework, we also introduce the generalized Legendre-Hermite Appell polynomials and derive their specific results. As special cases, the Legendre- Hermite-Bernoulli, Legendre-Hermite-Euler, and Legendre-Hermite-Genocchi polynomials are obtained, and their algebraic as well as operational features are analyzed. The findings presented herein enhance the theoretical development of special polynomial sequences and expand their potential applications in mathematical physics and differential equation analysis.