A bivariate approach to realrootedness of special polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866917860007215104 |
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| author | Gribinski, Aurelien Xavier |
| author_facet | Gribinski, Aurelien Xavier |
| contents | In this paper, we exhibit new monotonicity properties of roots of families of orthogonal polynomials $P_n^{(z)}(x)$ depending polynomially on a parameter (Laguerre and Gegenbauer). By establishing that $P_n^{(z)}(x)$ are realrooted in $z$ for $x$ in the support of orthogonality, we show realrootedness in $x$ and interlacing properties of $\partial_z^kP_n^{(z)}(x)$ for $k\leq n$ and $z \geq 0$, establishing a dual approach to orthogonality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00642 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A bivariate approach to realrootedness of special polynomials Gribinski, Aurelien Xavier Classical Analysis and ODEs In this paper, we exhibit new monotonicity properties of roots of families of orthogonal polynomials $P_n^{(z)}(x)$ depending polynomially on a parameter (Laguerre and Gegenbauer). By establishing that $P_n^{(z)}(x)$ are realrooted in $z$ for $x$ in the support of orthogonality, we show realrootedness in $x$ and interlacing properties of $\partial_z^kP_n^{(z)}(x)$ for $k\leq n$ and $z \geq 0$, establishing a dual approach to orthogonality. |
| title | A bivariate approach to realrootedness of special polynomials |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2301.00642 |