Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ satisfies a Painlevé ${\rm P}_{\rm IV}$ equation.
We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlevé equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre …
We investigate generalizations of the Charlier 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 solutions of the fifth Painlevé equation PV (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlevé equation. The first example deals with Freud weights $|x|^ρ\exp(-|x|^m)$ on the real line, and we repeat Freud's derivation and analysis for the cases $m=2,4,6$. The Freud equations for the recurrence coefficients when $m=4$ corresponds to the discrete Painlevé I equation. The second example deals with orthogonal polynomials on the unit circle for the weight $\exp(λ\cos θ)$. These orthogonal polynomials are important in the theory of random unitary matri…
We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlevé equations and the differential equation is the $σ$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.
We give a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlevé equations (discrete and continuous).
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.
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])…
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…
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 $α-q-P_V$.
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.
In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlevé transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the last part of this article, we list several open problems.
We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlevé equations, in a Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics and probability and many other disciplines. In these notes we give an introduction to the use of orthogonal polynomials in random matrix theory, we explain the notion of multiple orthogonal polynomials, and we show the link with certain non-linear difference and differential equations known as Painlevé 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…
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}…
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.
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…
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…
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.
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…
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…
Multiple orthogonal polynomials satisfy a number of recurrence relations, in particular there is a $(r+2)$-term recurrence relation connecting the type II multiple orthogonal polynomials near the diagonal (the so-called step-line recurrence relation) and there is a system of $r$ recurrence relations connecting the nearest neighbors (the so-called nearest neighbor recurrence relations). In this paper we deal with two problems. First we show how one can obtain the nearest neighbor recurrence coefficients (and in particular the recurrence coefficients of the orthogonal polynomials for each of the…
We study the orthogonal polynomials and the Hankel determinants associated with Gaussian weight with two jump discontinuities. When the degree $n$ is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected to the coupled Painlevé IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree $n$ grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlevé II system. As applications, we re-…
This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic fashion by introducing ladder operators that map an orthogonal polynomial to another by incrementing or decrementing its associated parameters by one.
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are asymptotics for the polynomials.
We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this technique is to provide a solution to a problem recently raised by M. E. H. Ismail.
In this paper we prove in a constructing way that exceptional Charlier, Meixner, Hermite and Laguerre polynomials satisfy higher order recurrence relations. Our conjecture is that the recurrence relations provided in this paper have minimal order.
We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + ν} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \to \infty$ with $N/n \to 1$ and $ν$ is a fixed number in $\mathbb{R} \setminus \mathbb{N}_0$. With the effect of the factor $(z-1)^{2b}$, the local parametrix near the critical point $z =1$ can be constructed in terms of $Ψ$-functions associated with the Painleve IV equation. We show that the asymptotics of the recurrence coefficients of orthogonal polynomials can be described in terms of specified solution of the Painleve IV equation in the double sca…
We show that multiple orthogonal polynomials for r measures $(μ_1,...,μ_r)$ satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices $\vec{n}\pm \vec{e}_j$, where $\vec{e}_j$ are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of measures $μ_j$. We show how the Christoffel-Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explic…
It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. In this paper we extend this result and show that every system of polynomials satisfying some $(2N+1)$-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are $N\times N$ matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner produ…
In this work is presented a study on matrix biorthogonal polynomials sequences that satisfy a nonsymmetric recurrence relation with unbounded coefficients. The ratio asymptotic for this family of matrix biorthogonal polynomials is derived in quite general assumptions. It is considered some illustrative examples.
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…
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…
In this paper, we derive new recurrence relations for two-variable orthogonal polynomials for example Jacobi polynomial, Bateman's polynomial and Legendre polynomial via two different differential operators $Ξ=\left(\frac{\partial }{\partial z} +\sqrt{w} \frac{\partial }{\partial w} \right)$ and $Δ=\left(\frac{1}{w} \frac{\partial }{\partial z} +\frac{1}{z} \frac{\partial }{\partial w} \right)$. We also derive some special cases of our main results.
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…
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…
We consider a sequence of polynomials $\{P_n\}_{n \geq 0}$ satisfying a special $R_{II}$ type recurrence relation where the zeros of $P_n$ are simple and lie on the real line. It turns out that the polynomial $P_n$, for any $n \geq 2$, is the characteristic polynomial of a simple $n \times n$ generalized eigenvalue problem. It is shown that with this $R_{II}$ type recurrence relation one can always associate a positive measure on the unit circle. The orthogonality property satisfied by $P_n$ with respect to this measure is also obtained. Finally, examples are given to justify the results.
We study the Hankel determinant of the generalized Jacobi weight $(x-t)^γx^α(1-x)^β$ for $x\in[0,1]$ with $α, β>0$, $t < 0 $ and $γ\in\mathbb{R}$. Based on the ladder operators for the corresponding monic orthogonal polynomials $P_n(x)$, it is shown that the logarithmic derivative of Hankel determinant is characterized by a $τ$-function for the Painlevé VI system.
We establish some new Turán's type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.
We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.
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…
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.
Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's theorem states that the converse is true. A general method to derive the coefficients $A_n$, $B_n$, $C_n$ in terms of the polynomial coefficients of the divided-difference equations satisfied by orthogonal polynomials on a quadratic or $q$-quadratic lattice is recalled. The Maple implementations rec2ortho of Koorwinder and Swarttouw or retode of Koepf and Schmers…
Zeilberger's algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations, and an adaption of Almquist and Zeilberger computes recurrence and differential equations for hyperexponential integrals. Further versions of this algorithm allow the computation of recurrence and differential equations from Rodrigues type formulas and from generating functions. In particular, these algorithms can be used to compute the differential/difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from th…
We study polynomials that are orthogonal with respect to a varying quartic weight \exp(-N(x^2/2+tx^4/4)) for t<0, where the orthogonality takes place on certain contours in the complex plane. Inspired by developments in 2D quantum gravity, Fokas, Its, and Kitaev, showed that there exists a critical value for t around which the asymptotics of the recurrence coefficients are described in terms of exactly specified solutions of the Painleve I equation. In this paper, we present an alternative and more direct proof of this result by means of the Deift/Zhou steepest descent analysis of the Riemann-…
We give the asymptotic behavior of the ratio of two neighboring multiple orthogonal polynomials under the condition that the recurrence coefficients in the nearest neighbor recurrence relations converge.
Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence of these singular points on the asymptotic behaviour of the recurrence coefficients is investigated.
