Truncated Hermite polynomials

We define the family of truncated Hermite polynomials $P_{n}left( x;zright) $, orthogonal with respect to the linear functional [Lleft[ pright] = int_{-z}^{z} pleft( xright) e^{-x^{2}} ,dx. ] The connection of $P_{n}left( x;zright) $ with the Hermite and Rys polynomials is stated. The semiclassical character of $P_{n}left( x;zright) $ as polynomials of class $2$ is emphasized. As a consequence, several properties of $P_{n}left( x;zright) $ concerning the coefficients $gamma_{n}left( zright) $ 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 the linear functional $L$, a holonomic differential equation (in $x)$ for the polynomials $P_{n}left( x;zright) $, and a nonlinear ODE for the functions $gamma_{n}left( zright) $ are deduced.