Truncated Hermite polynomials

Abstract

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

37 pages