A new family of two-variable polynomials based on hermite polynomials

Abstract

Abstract The aim of this paper is to introduce a new two-variable polynomials defined via Hermite polynomials. In order to construct some fundamental properties of these polynomials, we first derive a generating function relation. By using definition and this generating relation, we arrive at several recurrence relations, an integral representation, some implicit summation formulae, a symmetry identity for these new two-variable polynomials. Furthermore, we obtain some results which give various classes of multilinear and multilateral generating functions. Then, some special cases are presented. Finally, we also give a general class of these new polynomials and prove explicit closed-form formulae of them.