Zeros of linear combinations of Hermite polynomials
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918028824805376 |
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| author | Durán, Antonio J. |
| author_facet | Durán, Antonio J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of linear combinations of Hermite polynomials Durán, Antonio J. Classical Analysis and ODEs 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. |
| title | Zeros of linear combinations of Hermite polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2505.15330 |