Truncated Hermite polynomials

We define the family of truncated Hermite polynomials P-n (x; z), orthogonal with respect to the linear functional L [p] = integral (z) (-z) p (x) e(-x2) dx, p is an element of R[x], z > 0. The connection of Pn (x; z) with Hermite and Rys polynomials is stated. The semiclassical character of Pn (x; z) as polynomials of class 2 is emphasized. As a consequence, several properties of gamma n (x; z) concerning the coefficients.n (z) in the three-term recurrence relation they satisfy as well as the moments and the Stieltjes function of L are given. Ladder operators associated with such a linear functional and the holonomic equation that the polynomials Pn (x; z) satisfy are deduced.