Exploring the properties of multivariable Hermite polynomials in relation to Apostol-type Frobenius–Genocchi polynomials

This work presents a general framework that innovates and explores different mathematical aspects associated with special functions by utilizing the mathematical physics-based idea of monomiality. This study presents a unique family of multivariable Hermite polynomials that are closely related to Frobenius–Genocchi polynomials of Apostol type. The study’s deductions address the differential equation, generating expression, operational formalism, and other characteristics that define these polynomials. The affirmation of the controlling monomiality principle further confirms their mathematical foundations. In addition, the work proves recurrence relations, fractional operators, summation formulae, series representations, operational and symmetric identities, and so on, all of which contribute to our knowledge of these complex polynomials.