Yuanfei Lyu | Searcharxiv

arXiv2026

Hankel Determinant for a Perturbed Laguerre Weight with Pole Singularities and Generalized Painlevé III' Equation

We study the Hankel determinant for the weight $x^α{\rm exp}(-x-t_1/x-t_2/x^2), x\in[0,+\infty)$, with $α>-1,~t_1\in\mathbb{R}\setminus\{0\}, ~t_2>0.$ Compared with the weight $x^α{\rm e}^{-x-t_1/x}$ studied in prior work (where $α,t_1>0$), the range of $α$ in our work is extended and the parameter $t_2$ introduces a ``stronger" zero at the origin. This leads to more varied behavior of the Hankel determinant, and the interplay between $t_1$ and $t_2$ introduces uncertainty and complexity into the analysis. By using a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions, we show that the recurrence coefficients are expressed in terms of four auxiliary quantities which satisfy a system of difference equations that can be iterated. We also establish two coupled second order partial differential equations (PDEs) satisfied by two of the auxiliary quantities, which are reduced to a Painlevé III$^\prime$ equation when $t_2\rightarrow0^+$. Moreover, the logarithmic derivative of the Hankel determinant is shown to satisfy a second order six degree PDE which is reduced to the $σ$-form of the Painlevé III$^{\prime}$ equation when $t_2\rightarrow0^+$. Under suitable double scaling, we obtain the limiting forms of the above PDEs and deduce the equilibrium density of the eigenvalues for the unitary ensemble. We extend our analysis to the Hankel determinant for $x^α\exp(-x-\sum_{k=1}^m t_k/x^k)$ with $m=3$. For general $m$, we outline a derivation that leads, at least in principle, to the PDE satisfied by the logarithmic derivative of the Hankel determinant.

math-ph↗

arXiv2025

Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials

In the literature concerning the Laguerre-type weight function $x^λw_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^α(1+x)^βw_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function $x^α(1-x)^βw_0(x), x\in[0,1]$, with $w_0(x)$ continuously differentiable, the parameters $λ,α,β$ are usually constrained to be strictly positive to ensure the validity of the results. Recently, in [C. Min and P. Fang, Physica D 473 (2025), 134560 (9pp)], the ladder operators for the monic Laguerre-type orthogonal polynomials with $λ>-1$ were derived by exploiting the orthogonality properties. The quantities $A_n$ and $B_n$, which appear as coefficients in the ladder operators, exhibit different expressions compared with the previous ones for $λ>0$. In this paper, we construct an alternative deduction by making use of the Riemann-Hilbert problem satisfied by the orthogonal polynomials. Moreover, we employ both derivation strategies mentioned above to produce the ladder operators for the monic standard and shifted Jacobi-type orthogonal polynomials with $α,β>-1$. When $λ,α,β$ are restricted to positive values, our expressions of $A_n$ and $B_n$ are consistent with those in prior work. We present examples to validate our findings and generalize the existing conclusions, established by using the three compatibility conditions of the ladder operators and differentiating the orthogonality relations for the monic orthogonal polynomials, from $λ,α,β>0$ to $λ,α,β>-1$.

math.CA↗