Certain families of differential equations associated with the generalized 1-parameter Hermite–Frobenius Euler polynomials

This study introduces a new approach to the development of generalized 1-parameter, 2-variable Hermite–Frobenius–Euler polynomials, which are characterized by their generating functions, series definitions and summation formulae. Additionally, the research utilizes a factorization method to establish recurrence relations, shift operators and various differential equations, including differential, integro-differential and partial differential equations. The framework elucidates the fundamental properties of these polynomials by utilizing generating functions, series definitions and summation formulae. The results of the study contribute to the understanding of the properties of these polynomials and their potential applications.