On polynomials associated to product and composition of generating functions

Abstract

In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition $f\cdot (g\circ h)$ of generating functions $f$, $g$, and $h$. We give explicit formulae and apply the obtained result to two special families of polynomials; the first concerns the generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by $M.Mihoubi$ and $M.Sahari$. The second concerns two-variable Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by $N.U.Khan\mathit{\text{et al.}}$.