On a generalization of Hermite polynomials

We consider a new generalization of Hermite polynomials to the case of several variables. Our construction is based on an analysis of the generalized eigenvalue problem for the operator ∂Ax+D∂Ax+D, acting on a linear space of polynomials of N variables, where A   is an endomorphism of the Euclidean space RNRN and D is a second order differential operator. Our main results describe a basis for the space of Hermite–Jordan polynomials.