Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures
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arXiv
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| Format: | Preprint |
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2024
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| author | Kozhan, Rostyslav Vaktnäs, Marcus |
| author_facet | Kozhan, Rostyslav Vaktnäs, Marcus |
| contents | Given multiple orthogonal polynomials on the real line with respect to a system $\bmμ = (μ_1,\ldots,μ_r)$, we investigate multiple orthogonal polynomials associated with any rational perturbation of the form
$$
\widetilde{\bmμ}=\Big(\frac{Φ_1}{Ψ_{1}} μ_1,\dots,\frac{Φ_r}{Ψ_r}μ_r\Big),
$$ for any polynomials $Φ_1,\dots,Φ_r$ and $Ψ_1,\dots,Ψ_r$. We derive the analogues of Uvarov's determinantal formula for the multiple orthogonal polynomials of type I and type II for $\widetilde{\bmμ}$ and establish necessary and sufficient condition for normality of the indices.
The result allows the polynomials $\{Φ_j,Ψ_j\}_{j=1}^r$ to be arbitrary and permits the addition of finitely many point masses to each of the measures $μ_j$. Moreover, the measures $μ_j$ may be taken as quasi-definite linear functionals, which is of interest even in the case $r=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures Kozhan, Rostyslav Vaktnäs, Marcus Classical Analysis and ODEs 42C05, 41A21, 41A28 Given multiple orthogonal polynomials on the real line with respect to a system $\bmμ = (μ_1,\ldots,μ_r)$, we investigate multiple orthogonal polynomials associated with any rational perturbation of the form $$ \widetilde{\bmμ}=\Big(\frac{Φ_1}{Ψ_{1}} μ_1,\dots,\frac{Φ_r}{Ψ_r}μ_r\Big), $$ for any polynomials $Φ_1,\dots,Φ_r$ and $Ψ_1,\dots,Ψ_r$. We derive the analogues of Uvarov's determinantal formula for the multiple orthogonal polynomials of type I and type II for $\widetilde{\bmμ}$ and establish necessary and sufficient condition for normality of the indices. The result allows the polynomials $\{Φ_j,Ψ_j\}_{j=1}^r$ to be arbitrary and permits the addition of finitely many point masses to each of the measures $μ_j$. Moreover, the measures $μ_j$ may be taken as quasi-definite linear functionals, which is of interest even in the case $r=1$. |
| title | Determinantal Formulas for Rational Perturbations of Multiple Orthogonality Measures |
| topic | Classical Analysis and ODEs 42C05, 41A21, 41A28 |
| url | https://arxiv.org/abs/2407.13961 |