Certain Properties and Characterizations of Generalized Gould–Hopper-Based Hybrid Polynomials

This study offers a comprehensive generalization of the Gould–Hopper polynomials and their Appell-type analogs. Employing the quasi-monomiality approach, we delineate fundamental analytical characteristics, including recurrence relations, associated multiplicative and differential operators, and governing differential equations. Additionally, we derive series representations and determinantal expressions for this newly defined polynomial family. Within this framework, several significant subclasses are introduced and examined, such as the generalized Gould–Hopper-based Appell polynomials. The formulation is further extended using fractional operator techniques to explore their intrinsic structural attributes. Moreover, we construct and investigate new families, namely, the generalized Gould–Hopper-based Bernoulli, Gould–Hopper-based Euler, and Gould–Hopper-based Genocchi polynomials, emphasizing their operational and algebraic properties. Collectively, these findings advance the theory of special functions and provide a foundation for potential applications in mathematical physics and the study of differential equations.