Zeros of linear combinations of Hermite polynomials

TL;DR

This paper investigates the real zeros of finite linear combinations of consecutive Hermite polynomials, revealing that the zeros' reality depends on the roots of an associated polynomial, with implications for understanding polynomial zero distributions.

Contribution

It establishes a connection between the zeros of linear combinations of Hermite polynomials and the roots of an associated polynomial, providing conditions for all zeros to be real.

Findings

If all roots of the polynomial P are real, then all zeros of q_n are real for n ≥ K.

The polynomial P(x) determines the zero distribution of the linear combination.

The study considers two normalizations of Hermite polynomials, standard and Appell, with similar zero properties.

Abstract

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde H_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\dots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$. We consider two different normalizations of Hermite polynomials: the standard one (i.e. $\tilde H_n=H_n$), and $\tilde H_n=H_n/(2^nn!)$ (so that $q_n$ are Appell polynomials: $q_n'=q_{n-1}$). In both cases, we show the key role played by the polynomial $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$ to solve this problem. In particular, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real.