Second order exponential differential operator and generalized Hermite polynomials

Both univariate and multivariate generalized Hermite polynomials are defined by introducing second order exponential differential operator, corresponding probabilistic expressions are given explicitly. Based on those expectation expressions, their generating functions, recursion relations and other properties are derived conveniently in connection to Stein's identities. In particular, orthogonality are investigated and associated expressions are presented accordingly. Additionally, some other classical identities related to Hermite polynomials are proved by an alternative way, and some examples and remarks are presented.