Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert–Hermite polynomials via generalized Fibonacci polynomials

Abstract

In this paper, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain a new explicit formula for the generalized Humbert–Hermite polynomials. We provide determinantal representations for the ratio of two differentiable functions. We obtain a recursive relation for the generalized Humbert–Hermite polynomials. As a practice, we derive an alternative recursive relation for generalized Humbert–Hermite polynomials via the Hessenberg determinant. Finally, we derive several families of multilinear and multilateral generating functions for the generalized Humbert–Hermite‐type polynomials and other polynomials which are mentioned in this paper. Our results also include many well‐known polynomials in the literature.