Truncated Hermite polynomials

TL;DR

This paper studies a family of truncated Hermite polynomials, exploring their recurrence relations, differential equations, and electrostatic properties of their zeros, providing new insights into their mathematical structure.

Contribution

It introduces and analyzes the properties of truncated Hermite polynomials, including their recurrence coefficients, differential equations, and zero distribution, which are novel extensions of classical Hermite polynomial theory.

Findings

Coefficients in the 3-term recurrence relation are characterized.

A second order differential equation for the polynomials is derived.

Zeros of the polynomials have an electrostatic interpretation.

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.