Zeros of linear combinations of Hermite polynomials

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$q_n(x)=\sum _{j=0}^K{\gamma }_j{\tilde{H}}_{n-j}(x),n\ge K,$$ where ${\gamma }_j$, $j=0,\dots ,K$, are real numbers with ${\gamma }_0=1$, ${\gamma }_K≠0$. We consider two different normalizations of Hermite polynomials: the standard one (i.e. ${\tilde{H}}_n=H_n$), and ${\tilde{H}}_n=H_n\mathrm{/}(2^{n}n!)$ (so that $q_n$ are Appell polynomials: $q_n^{\mathrm{\prime }}=q_{n-1}$). In both cases, we show the key role played by the polynomial $P(x)={\sum }_{j=0}^K{\gamma }_j{x}^{K-j}$ 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.