Zeros of linear combinations of Hermite polynomials
Antonio J. Durán
TL;DR
The paper analyzes the real-rootedness of finite linear combinations of consecutive Hermite polynomials under two normalizations and reveals that the zeros are governed by the auxiliary polynomial $P(x)=\sum_{j=0}^K\gamma_j x^{K-j}$. By leveraging the backward shift operator and the framework of generalized Hermite polynomials, the authors establish precise conditions for when all zeros are real and simple and when interlacing with successive polynomials holds, with distinct behavior depending on whether $P$ has only real zeros or also nonreal zeros. They further extend the analysis to the Appell normalization, obtaining generating-function-based asymptotics and Turán-type inequalities, and provide Mehler–Heine type asymptotics and detailed zero-counting results that describe the distribution of real and nonreal zeros as $n\to\infty$. The work connects to type II multiple Hermite polynomials and yields a comprehensive picture of how linear combinations of Hermite polynomials behave spectrally, including zero distribution and monotonicity properties with respect to the coefficients. These results have implications for spectral theory and the study of zero distributions in classical orthogonal polynomial families.
Abstract
We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_j\tilde H_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\dots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$. 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 $P(x)=\sum_{j=0}^Kγ_jx^{K-j}$ 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.