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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. 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Dur\\'an","submitted_at":"2025-05-21T10:04:08Z","abstract_excerpt":"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. 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