Some generating functions and other properties associated with the generalized Hermite and related polynomials

Abstract

In this article, we first introduce and study a new family of the generalized Hermite polynomials , which are an extension of the generalized Hermite polynomials {h n, m (x)} [2,6]. We next consider the polynomials and , which (just as the generalized Hermite polynomials {h n, m (x)}) are seemingly interesting variants of the familiar Gould–Hopper polynomials [4,6]. For the general class of polynomials , we derive several generating functions and find an explicit formula in terms of the Srivastava–Daoust multivariable hypergeometric functions. Several properties are derived also for the polynomials and . The results presented here are motivated essentially by those that were derived in our earlier work [3].