Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912690419531776 |
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| author | Chen, Siqi Zhu, Mengkun |
| author_facet | Chen, Siqi Zhu, Mengkun |
| contents | In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^λ\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $λ>-1$, $s\in\mathbb{R}$, and Painlevé equations. Specifically, we introduce a new transformation for the expressions obtained by Filipuk et al. in their analysis of ladder operators for semiclassical Laguerre polynomials, thereby deriving a recurrence relation. Subsequently, we establish a correspondence between this recurrence relation and a class of d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations Chen, Siqi Zhu, Mengkun Classical Analysis and ODEs In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^λ\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $λ>-1$, $s\in\mathbb{R}$, and Painlevé equations. Specifically, we introduce a new transformation for the expressions obtained by Filipuk et al. in their analysis of ladder operators for semiclassical Laguerre polynomials, thereby deriving a recurrence relation. Subsequently, we establish a correspondence between this recurrence relation and a class of d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations. |
| title | Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.04168 |