Recurrence coefficients for the semiclassical Laguerrre weight and d-P$\left(A_2^{(1)}\mathrm{/}{E}_6^{(1)}\right)$ equations

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 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$\left(A_2^{(1)}\mathrm{/}{E}_6^{(1)}\right)$ equations.