Bessel Type Orthogonality For Hermite Polynomials

TL;DR

This paper demonstrates a Bessel type orthogonality relation for Hermite polynomials involving their zeros and a finite interval, extending applicability to polynomials of degree three and higher.

Contribution

It introduces a novel Bessel type orthogonality relation for Hermite polynomials based on their zeros, broadening the understanding of their orthogonal properties.

Findings

Hermite polynomials satisfy a Bessel type orthogonality relation.

The relation involves zeros of a single index Hermite polynomial.

Applicability extends to Hermite polynomials with degree n ≥ 3.

Abstract

It is shown that Hermite polynomials satisfy a Bessel type orthogonality relation, based on the zeros of a single index Hermite polynomial and with a finite integration interval. Because of the role of non-symmetric zeros in the final relation, its applicability covers Hermite polynomials $P_n(x)$ with $n\ge 3$.