Zeros of linear combinations of Laguerre polynomials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911083646681088 |
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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 Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_j\tilde L^α_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^α$, the polynomials $\mathcal L_n^α=n!L_n^α/(1+α)_n$ (so that $\mathcal L_n^α(0)=1$), the standard Laguerre polynomials $(L_n^α)_n$ and the Brenke normalization $L_n^α/(1+α)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^jγ_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^Kγ_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $α+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22425 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of linear combinations of Laguerre polynomials Durán, Antonio J. Classical Analysis and ODEs We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^Kγ_j\tilde L^α_{n-j}(x),\quad n\ge K, $$ where $γ_j$, $j=0,\cdots ,K$, are real numbers with $γ_0=1$, $γ_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^α$, the polynomials $\mathcal L_n^α=n!L_n^α/(1+α)_n$ (so that $\mathcal L_n^α(0)=1$), the standard Laguerre polynomials $(L_n^α)_n$ and the Brenke normalization $L_n^α/(1+α)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^jγ_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^Kγ_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $α+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough. |
| title | Zeros of linear combinations of Laguerre polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2507.22425 |