Painlevé equations for semi-classical recurrence coefficients

Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.

Preud's equations for orthogonal polynomials as discrete Painlevé equations

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in order to study special continued fractions, recurrence relations, and various asymptotic expansions (G. Freud's contribution [28, 56]). Most striking example is n = 2tw_n + w_n(w_n+1 + w_n + w_n-1) for the recurrence coefficients p_n+1 = xp_n - w_np_n-1 of the orthogonal polynomials related to the weight w(x) = exp(-4(tx^3 + x^4)) (notation of [26, pp. 34-36])…

$q$-Discrete Painlevé equations for recurrence coefficients of modified $q$-Freud orthogonal polynomials

We present an asymmetric $q$-Painlevé equation. We will derive this using $q$-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this $q$-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution which gives the recurrence coefficients. We establish a connection with $α-q-P_V$.

Variations of Stieltjes-Wigert and q-Laguerre polynomials and their recurrence coefficients

We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlevé III equation. Next we consider the q-Laguerre or generalized Stieltjes-Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlevé V eq…

Asymptotics of recurrence coefficients for the Laguerre weight with a singularity at the edge

In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the Deift-Zhou steepest descent method. The asymptotic formulas of the leading coefficients and the recurrence coefficients for large n are described in terms of a class of analytic solutions to the the σ-form of the Painlevé II equation and the Painlevé XXXIV equation.

Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and N-recurrences for averages over U(N) -- \PIIIa and \PV $τ$-functions

In this work we show that the $ N\times N $ Toeplitz determinants with the symbols $ z^μ\exp(-{1/2}\sqrt{t}(z+1/z)) $ and $ (1+z)^μ(1+1/z)^ν\exp(tz) $ -- known $τ$-functions for the \PIIIa and \PV systems -- are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlevé equations associated with the degenerations of the rational surfaces $ D^{(1)}_{6} \to E^{(1)}_{7} $ (discrete Painlevé {\rm II}) and $ D^{(1)}_{5} \to E^{(1)}_{6}…

The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations

Over the last decade it has become clear that discrete Painlevé equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlevé equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answer…

Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlevé equations via the geometric approach

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlevé equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlevé equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrenc…

The generalized Krawtchouk polynomials and the fifth Painlevé equation

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.

Painlevé I double scaling limit in the cubic random matrix model

We obtain the double scaling asymptotic behavior of the recurrence coefficients and the partition function at the critical point of the $N\times N$ Hermitian random matrix model with cubic potential. We prove that the recurrence coefficients admit an asymptotic expansion in powers of $N^{-2/5}$, and in the leading order the asymptotic behavior of the recurrence coefficients is given by a Boutroux tronquée solution to the Painlevé I equation. We also obtain the double scaling limit of the partition function, and we prove that the poles of the tronquée solution are limits of zeros of the partiti…

Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For $t>0,$ the factor $\rex^{-t/x}$ induces an infinitely strong zero at $x=0.$ With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a syst…

Painlevé IV、Chazy II 以及半经典 Laguerre 多项式递推系数及其 Hankel 行列式的渐近性

Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants

本文研究首一半经典 Laguerre 多项式，证明非对角递推系数满足 Chazy II，并结合库仑流体与离散系统得到递推系数、次首项系数及 Hankel 行列式的大 n 渐近展开。

Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight $w(x)=x^αe^{-x-t/x}$, $x\in (0, \infty)$, $t>0$ and $α>0$. When the matrix size $n\to\infty$, we obtain an asymptotic formula for the Hankel determinants, valid uniformly for $t\in (0, d]$, $d>0$ fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large-$n$ asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier pa…

A Generalized Freud Weight

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2λ+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $λ>-1$ and $t\in\mathbb{R}$, and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation. Further we derive a second-order linear ordinary differential equatio…

Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex plane. When the parameters $n$, $t$ and the degree $k$ are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy $τ$-function of the Painlevé III equation. When the degree $k=n$, $n$ is large and $t$ is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients…

Discrete Painlevé equations, Orthogonal Polynomials on the Unit Circle and $N$-recurrences for averages over U(N) -- \PVI $τ$-functions

The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of orthogonal polynomials and associated functions. In particular for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{ρ_j} $, consisting…

Multiple orthogonal polynomials associated with an exponential cubic weight

We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and nearest-neighbor recurrence relations. It turns out that the recurrence coefficients are related to a discrete Painlevé equation. The asymptotics of the recurrence coefficients, the ratio of the diagonal multiple orthogonal polynomials and the (scaled) zeros of these polynomials are also investigated.

第三类Painlevé方程的Umemura多项式的构造性证明

A Constructive Proof for the Umemura Polynomials of the Third Painlevé Equation

本文扩展Taneda方法，为第三类Painlevé方程的Umemura多项式提供构造性代数证明，证实非线性递推生成的有理函数实为多项式，并揭示其根的信息。

Classical Solutions of the Degenerate Fifth Painlevé Equation

In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to expressible in terms of the third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.

Orthogonal polynomials on a bi-lattice

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-β. We combine both lattices to obtain the bi-lattice N \cup (N+1-β) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlevé equation.

带跳跃的高斯权正交多项式与离散潘勒韦方程

Orthogonal Polynomials for the Gaussian Weight with a Jump and Discrete Painlevé Equations

本文提出精细化的离散潘勒韦识别问题，强调超越表面类型描述动力学，并以带单跳跃高斯权正交多项式中的递推关系为例展示求解方法。

Matrix Jacobi Biorthogonal Polynomials via Riemann-Hilbert problem

We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential relations that these matrix orthogonal polynomials and the second kind functions associated to them verify. For the corresponding matrix recurrence coefficients, non-Abelian extensions of a family of discrete Painlevé d-PIV equations are obtained for the three term recurrence relation coefficients.

与Krawtchouk多项式相关的微分系统：迭代正则化与Painlevé方程

Differential system related to Krawtchouk polynomials: iterated regularisation and Painlevé equation

研究通过迭代正则化将广义Krawtchouk多项式相关微分系统与Painlevé V方程联系起来，并探讨正则化如何产生多项式系统及birational变换的分解。

正交多项式：从Heun方程到Painlevé方程

Orthogonal polynomials: from Heun equations to Painlevé equations

本文研究了四种权函数对应的正交多项式，给出了其满足的二阶线性微分方程及Heun方程，利用等单值形变方法将其转化为Painlevé方程，并讨论了Hankel行列式的渐近行为。

Asymptotics of Integrals of Some Functions Related to the Degenerate Third Painlevé Equation

It is shown how to calculate asymptotics of integrals over the positive semi-axis of two functions related to the Degenerate Third Painlevé Equation (dP3). As an example, the corresponding results for the meromorphic solution of the dP3 vanishing at the origin are presented.

半经典 Laguerre 权重的递推系数与 d-P$\left(A_{2}^{(1)}/E_{6}^{(1)}\right)$ 方程

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

本文借助 Sakai 几何框架研究半经典 Laguerre 权重递推系数与 Painlevé 方程的联系，通过新变换导出递推关系，并建立其与 d-P(A2(1)/E6(1)) 方程的对应。

Painlevé V and time dependent Jacobi polynomials

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $α_n$ and $β_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a partic…

非多项式q-Askey方案的半经典极限

Semiclassical limit of a non-polynomial $q$-Askey scheme

本文证明了非多项式双曲q-Askey方案底部元素M和Q的半经典渐近公式，并揭示其指数是Painlevé I和III₃方程单值流形上Darboux坐标对之间典范变换的生成函数，进而猜想该方案其他成员对应其他Painlevé方程。

Krall型正交多项式与可积等单调变形

Krall-type orthogonal polynomials and integrable isomonodromic deformations

本文研究了Krall型正交多项式与可积等单调变形的关系，揭示了其与Painlevé方程及Schlesinger型偏微分方程的联系。

Painlevé IV方程、双退化Heun方程与由不连续半经典Laguerre权生成的Hankel行列式

Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight

研究不连续半经典Laguerre权函数，推导出与Painlevé IV方程相关的递推系数，并揭示正交多项式满足双退化Heun方程。

Recurrence Coefficients of a New Generalization of the Meixner Polynomials

We investigate new generalizations of the Meixner polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-β$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-β)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlevé equation P$_{\textup V}$. Initial conditions for different lattices can be transformed to the classical solutions of P$_{\textup V}$ with special values of the parameters. We also study one property of the Bäcklund transformation of P$_{\textup V}$.

Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyze relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlevé equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmui…

Unique positive solutions to $q$-discrete equations associated with orthogonal polynomials

Boelen et al. (2010) deduced a $q$-discrete Painlevé equation satisfied by the recurrence coefficients of orthogonal polynomials and conjectured that the equation had a unique positive solution. We prove their conjecture and discuss related work in the area, highlighting a potentially unifying methodology.

The diagonal two-point correlations of the Ising model on the anisotropic triangular lattice and Garnier systems

The diagonal spin-spin correlations $ \langle σ_{0,0}σ_{N,N} \rangle $ of the Ising model on a triangular lattice with general couplings in the three directions are evaluated in terms of a solution to a three-variable extension of the sixth Painlevé system, namely a Garnier system. This identification, which is accomplished using the theory of bi-orthogonal polynomials on the unit circle with regular semi-classical weights, has an additional consequence whereby the correlations are characterised by a simple system of coupled, nonlinear recurrence relations in the spin separation $ N \in \mathb…

The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type

Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270-297, arXiv:0808.3590] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Lague…

Discrete Painlevé equations for a class of \PVI $τ$-functions given as U(N) averages

In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $ w(z)=\prod^m_{j=1}(z-z_j(t))^{ρ_j} $ were derived.Here it is shown that in the case $ m=3 $ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev…

An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)

We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.

Bi-orthogonal systems on the unit circle, regular semi-classical weights and the discrete Garnier equations

We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrence relations of the discrete Painlevé type, for example in the case with one free deformation variable the system was found to be characterised by a solution to the discrete fifth Painlevé equation. Here …

延迟 Painlevé-I 方程、关联多项式与 Masur-Veech 体积

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

本文研究由 Shabat 穿衣链延迟周期约化得到的延迟 Painlevé-I 方程，构造其形式整函数解并引入 Bernoulli-Catalan 多项式，用 $A_1^{(1)}$ 仿射 Weyl 群刻画奇点结构，并重新推导 Masur-Veech 体积公式。

On the recurrence coefficients of generalized little $q$-Laguerre polynomials

In this paper we consider a semi-classical variation of the weight related to the little $q$-Laguerre polynomials and obtain a second order second degree discrete equation for the recurrence coefficients in the three-term recurrence relation.

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlevé Equation on Segments on Non-Positive Semi-Axis

We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$.

广义Freud权、离散Painlevé I hierarchy与Hankel行列式的完全渐近展开

Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants

本文研究广义Freud权对应的正交多项式与Hankel行列式，揭示离散Painlevé I hierarchy的统一结构，推导相关微分方程与恒等式，得到这些量的完全渐近展开并以十次Freud权为例说明。

A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential

We study the hermitian one matrix model with semi-classical potential. This is a general unitary invariant random matrix ensemble in which the potential has a derivative that is a rational function and the measure is supported on some collection of disjoint closed intervals. Such models have attracted much interest both due to their physical applications and relations to integrable systems. An object of central interest in random matrix theory is the correlation kernel, as this encodes the eigenvalue correlation functions. In recent years many results have been obtained proving that the correl…

广义Gauss-Rys正交多项式

Generalized Gauss-Rys orthogonal polynomials

本研究针对对称线性泛函下的首一正交多项式序列$(P_n(x;z;λ))_{n\geq 0}$，结合其三项递推关系的参数特性，分析广义Gauss-Rys正交多项式的性质，给出递推系数渐近展开、关联的离散Painlevé与Painlevé方程，以及零点的静电诠释和参数动力学。

Quantum Curve and the First Painlevé Equation

We show that the topological recursion for the (semi-classical) spectral curve of the first Painlevé equation $P_{\rm I}$ gives a WKB solution for the isomonodromy problem for $P_{\rm I}$. In other words, the isomonodromy system is a quantum curve in the sense of [Dumitrescu O., Mulase M., Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022] and [Dumitrescu O., Mulase M., arXiv:1411.1023].

关于离散Painlevé等价问题、非共轭变换和节点曲线

On the discrete Painlevé equivalence problem, non-conjugate translations and nodal curves

研究非自治差分方程系统与Laguerre和Meixner权重的广义化，识别为离散Painlevé方程，并探讨其在Sakai分类中的类型及非共轭变换和节点曲线的影响。

广义高阶 Freud 权函数

Generalised higher-order Freud weights

本文研究半经典广义高阶 Freud 权函数下的正交多项式，给出第一矩的 _1F_m 有限分拆和表示，证明递推系数属于第一离散 Painlevé 层级，并分析其渐近、零点分布及递推结构。

Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants

In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;α>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of $β_n(t)$ can be expressed in terms of a particular Painlevé V transcendent. The large $n$ asymptotic expansi…

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $π_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12 z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$, $N\to\infty$. Our goal is to describe these asymptotic behaviors globally for $t\in {\mathbb C}$ in different regions. We also describe the "breaking" curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic "phase portrait" of the recurrence coefficients of $π_n(z)$, which was studied num…

对称六次Freud权的系数序列性质研究

Symmetric Sextic Freud Weight

本文研究对称六次Freud权的递推系数序列性质，揭示其与可积系统的深刻联系。

Semi-classical Laguerre polynomials and a third order discrete integrable equation

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.