Orthogonal polynomials: from Heun equations to Painlevé equations
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929640171372544 |
|---|---|
| author | Zhu, Mengkun Chen, Yuting Yu, Jianduo Li, Chuanzhong |
| author_facet | Zhu, Mengkun Chen, Yuting Yu, Jianduo Li, Chuanzhong |
| contents | In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Dereziński, A. Ishkhanyan, A. Latosiński, SIGMA 17 (2021), 056], we transform these Heun equations into Painlevé equations. It is interesting that the Painlevé equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orthogonal polynomials: from Heun equations to Painlevé equations Zhu, Mengkun Chen, Yuting Yu, Jianduo Li, Chuanzhong Classical Analysis and ODEs In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Dereziński, A. Ishkhanyan, A. Latosiński, SIGMA 17 (2021), 056], we transform these Heun equations into Painlevé equations. It is interesting that the Painlevé equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.} |
| title | Orthogonal polynomials: from Heun equations to Painlevé equations |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2412.14604 |