Certain properties and applications of Hermite–Fubini–Appell polynomial systems in mathematical physics

Abstract In this paper, we introduce and investigate a new hybrid class of special polynomials, called the multivariate Hermite–Fubini based Appell polynomials of order α . The proposed family is constructed by integrating the combinatorial generating mechanism of Fubini polynomials of order α with the operational structure of three-variable Hermite–Appell polynomials within the framework of the monomiality principle. An explicit generating function for the new polynomial sequence is established, and several fundamental properties are derived. In particular, the operational identities, differential relations, and shift formulas for the multiplicative and derivative operators are derived. Moreover, a determinant representation of the polynomial family using Cramer’s rule is also obtained. Various special cases are also presented to demonstrate how the new polynomials are related to some well-known classical sequences. Theoretical findings are also supported by computational examples and graphical representations to describe the structural properties of the polynomials.