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 qₙ(x)=∑ⱼ₌₀Kγⱼ Hₙ₋ⱼ(x), n≥ K, where γⱼ, j=0, ,K, are real numbers with γ₀=1, γK =0. We consider two different normalizations of Hermite polynomials: the standard one (i.e. Hₙ=Hₙ), and Hₙ=Hₙ/(2ⁿn!) (so that qₙ are Appell polynomials: qₙ'=qₙ₋₁). In both cases, we show the key role played by the polynomial P(x)=∑ⱼ₌₀KγⱼxK-j to solve this problem. In particular, if all the zeros of P are real then all the zeros of qₙ, n≥ K, are also real.