Truncated Hermite polynomials

We consider the family of polynomials $p_n(x;z),$ orthogonal with respect to the inner product $$⟨f,g⟩={\int }_{-z}^{z}f\left(x\right)g\left(x\right)e^{-x^2}dx\mathrm{.}$$ We show some properties about the coefficients in their 3-term recurrence relation, connections between $p_n(x;z)$ and $p_n^{\mathrm{\prime }}(x;z),$ a second order differential equation satisfied by $p_n(x;z),$ and an electrostatic interpretation of their zeros.