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arXiv · 2511.04168

Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations

Siqi Chen·Mengkun Zhu
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Abstract

In this paper, we use Sakai's geometric framework to explore the profound interconnection between recurrence coefficients of the semiclassical Laguerre weight $w(x)=x^{\lambda}\mathrm{e}^{-x^2+sx}$, $x\in\mathbb{R}^+$, $\lambda>-1$, $s\in\mathbb{R}$, and Painlev\'e 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.

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Siqi Chen, Mengkun Zhu. 2025-11-06. Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ equations. https://arxiv.org/abs/2511.04168

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