  We consider a generalization of the classical Hermite polynomials by the
addition of terms involving derivatives in the inner product. This type of
generalization has been studied in the literature from the point of view of the
algebraic properties. Thus, our aim is to study the asymptotics of this
sequence of nonstandard orthogonal polynomials. In fact, we obtain
Mehler--Heine type formulas for these polynomials and, as a consequence, we
prove that there exists an acceleration of the convergence of the smallest
positive zeros of these generalized Hermite polynomials towards the origin.
