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The relationship between semi-classical Laguerre polynomials and the fourth Painlev\'e 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 Painlev\'e 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 Painlev\'e equation.


Introduction
In this paper we are concerned with the coefficients in the three-term recurrence relations for orthogonal polynomials with respect to the semi-classical Laguerre weight ω(x; t) = x λ exp(−x 2 + tx), x ∈ R + , (1.1) with parameters λ > −1 and t ∈ R, which has been recently studied by Boelen and van Assche [10] and Filipuk, van Assche and Zhang [28]. It is shown that these recurrence coefficients can be expressed in terms of Wronskians that arise in the description of special function solutions of the fourth Painlevé equation (P IV ) where A and B are constants, which are expressed in terms of parabolic cylinder functions. The relationship between semi-classical orthogonal polynomials and integrable equations dates back to the work of Shohat [67] and later Freud [39], as well as Bonan and Nevai [11]. However it was not until the work of Fokas, Its and Kitaev [31,32] that these equations were identified as discrete Painlevé equations. The relationship between semi-classical orthogonal polynomials and the (continuous) Painlevé equations was demonstrated by Magnus [53,54] who showed that the coefficients in the three-term recurrence relation for the Freud weight [11,39,73] ω(x; t) = exp − 1 4 x 4 − tx 2 , x ∈ R, with t ∈ R a parameter, can be expressed in terms of solutions of P IV (1.2).
A motivation for this work is the fact that recurrence coefficients of semi-classical orthogonal polynomials can often be expressed in terms of solutions of the Painlevé equations. For example, recurrence coefficients are expressed in terms of solutions of P II for semi-classical orthogonal polynomials with respect to the Airy weight ω(x; t) = exp 1 3 x 3 + tx , with t ∈ R a parameter [53]; in terms of solutions of P III for the perturbed Laguerre weight ω(x; t) = x α exp (−x − t/x) , x ∈ R + , with α > 0 and t ∈ R + parameters [18]; in terms of solutions of P V for the weights x ∈ R + , with α, β > 0 and t ∈ R + parameters [3,4,15,19,38]; and in terms of solutions of P VI for the generalized Jacobi weight with α, β, γ > 0 and t ∈ R + parameters [5,19,25,53].
Recurrence coefficients for orthogonal polynomials with respect to discontinuous weights which involve the Heaviside function H(x) have also been expressed in terms of solutions of Painlevé equations [2,20,35,37], while recurrence coefficients for orthogonal polynomials with respect to discrete weights have been expressed in terms of solutions of Painlevé equations [9,8,23,26,27].
This paper is organized as follows: in §2, we review some properties of orthogonal polynomials; in §3, we review some properties of the fourth Painlevé equation (1.2), including its Hamiltonian structure §3.1, Bäcklund and Schlesinger transformations §3.2 and special function solutions §3.3; in §4 we express the coefficients which arise in the three-term recurrence relation associated with orthogonal polynomials for the semi-classical Laguerre weight (1.1) in terms of Wronskians that arise in the description of special function solutions of P IV (1.2); in §5 we derive asymptotic expansions for the recurrence coefficients; in §6 we discuss orthogonal polynomials with respect to the semi-classical Hermite weight ω(x; t) = |x| λ exp(−x 2 + tx), x, t ∈ R, λ > −1, which is an extension of the semi-classical Laguerre weight (1.1) to the whole real line, and show that the recurrence coefficients are also expressed in terms of Wronskians that arise in the description of special function solutions of P IV (1.2); and in §7 we discuss our results.

Orthogonal polynomials
Let P n (x), n ∈ N, be the monic orthogonal polynomial of degree n in x with respect to a positive weight ω(x) on where δ m,n denotes the Kronekar delta. One of the most important properties of orthogonal polynomials is that they satisfy a three-term recurrence relationship of the form where the coefficients α n and β n are given by the integrals with P −1 (x) = 0 and P 0 (x) = 1. These coefficients in the three-term recurrence relationship can also be expressed in terms of determinants whose entries are given in terms of the moments associated with the weight ω(x). Specifically, the coefficients α n and β n in the recurrence relation (2.1) are given by with ∆ 0 = 1, ∆ −1 = 0, and ∆ n is the determinant with ∆ 0 = 0 and µ k , the kth moment, is given by the integral We remark that the Hankel determinant ∆ n (2.3a) also has the integral representation The monic polynomial P n (x) can be uniquely expressed as the determinant , and the normalisation constants as For further information about orthogonal polynomials see, for example [21,47,70]. Now suppose that the weight has the form where t is a parameter, with finite moments for all t ∈ R, which is the case for the semi-classical Laguerre weight (1.1). If the weight has the form (2.7), which depends on the parameter t, then the orthogonal polynomials P n (x), the recurrence coefficients α n , β n given by (2.2), the determinants ∆ n , ∆ n given by (2.3) and the moments µ k given by (2.4) are now functions of t. Specifically, in this case then Further, the recurrence relation has the form where we have explicitly indicated that the coefficients α n (t) and β n (t) depend on t.
Theorem 2.1. If the weight has the form (2.7), then the determinants ∆ n (t) and ∆ n (t) given by (2.3) can be written as

9)
where W(ϕ 1 , ϕ 2 , . . . , ϕ n ) is the Wronskian given by Proof. Since µ k = d k µ 0 dt k , the determinant ∆ n (t) can be written in the form as required, and the determinant ∆ n (t), can be written in the form as required.
The Hankel determinant ∆ n (t) satisfies the Toda equation, as shown in the following theorem.
Using Theorems 2.1 and 2.2 we can express the recurrence coefficients α n (t) and β n (t) in terms of derivatives of the Hankel determinant ∆ n (t) and so obtain explicit expressions for these coefficients. Theorem 2.3. The coefficients α n (t) and β n (t) in the recurrence relation (2.8) associated with monic polynomials orthogonal with respect to a weight of the form (2.7) are given by with ∆ n (t) is the Hankel determinant given by (2.9).
Proof. By definition the coefficients α n (t) and β n (t) in the recurrence relation (2.8) are given by where the determinants ∆ n and ∆ n are given by (2.3). Hence from (2.9) as required. By definition , and so from Theorem 2.2 we have β n (t) = d 2 dt 2 ln ∆ n (t), as required. See also Chen, Ismail and van Assche [17] who also discuss applications to random matrices.
Equivalently the recurrence coefficients α n (t) and β n (t) can be expressed in terms of h n (t) given by (2.6).
Lemma 2.4. The coefficients α n (t) and β n (t) in the recurrence relation (2.8) associated with monic polynomials orthogonal with respect to a weight of the form (2.7) are given by where h n (t) is given by (2.6).
Additionally the coefficients α n (t) and β n (t) in the recurrence relation (2.8) satisfy a Toda system.
Theorem 2.5. The coefficients α n (t) and β n (t) in the recurrence relation (2.8) associated with a weight of the form (2.7) satisfy the Toda system Proof. See Chen and Ismail [16], Ismail [47, §2.8, p. 41] and Moser [57]; see also [8] for further details and a direct proof in the case of a semi-classical weight of the form (2.7).
Suppose P n (x), for n ∈ N, is a sequence of classical orthogonal polynomials (such as Hermite, Laguerre and Jacobi polynomials), then P n (x) is a solution of a second-order ordinary differential equation of the form where σ(x) is a monic polynomial with deg(σ) ≤ 2, τ (x) is a polynomial with deg(τ ) = 1, and λ n is a real number which depends on the degree of the polynomial solution, see Bochner [7]. Equivalently, the weights of classical orthogonal polynomials satisfy a first-order ordinary differential equation, the Pearson equation with σ(x) and τ (x) the same polynomials as in (2.12), see, for example [1,7,21]. However for semi-classical orthogonal polynomials, the weight function ω(x) satisfies the Pearson equation (2.13) with either deg(σ) > 2 or deg(τ ) > 1, see, for example [45,55]. For example, the Pearson equation (2.13) is satisfied for the weight (1.1) with and so the weight (1.1) is indeed a semi-classical weight function. Filipuk, van Assche and Zhang [28] comment that "We note that for classical orthogonal polynomials (Hermite, Laguerre, Jacobi) one knows these recurrence coefficients explicitly in contrast to non-classical weights".
In §4 we show that, in the case of the semi-classical Laguerre weight (1.1), the determinants ∆ n (t) and ∆ n (t) can be explicitly written as Wronskians which arise in the description of special function solutions of P IV (1.2) that are expressed in terms of parabolic cylinder functions D ν (z) when λ ∈ Z, or error functions erf(z) when λ = n ∈ Z. Consequently the recurrence coefficients α n (t) and β n (t) (2.2) associated with orthogonal polynomials for the semiclassical Laguerre weight (1.1) can also be explicitly written in terms of these Wronskians.

Properties of the fourth Painlevé equation
The six Painlevé equations (P I -P VI ) were first discovered by Painlevé, Gambier and their colleagues in an investigation of which second order ordinary differential equations of the form where F is rational in dq/dz and q and analytic in z, have the property that their solutions have no movable branch points. They showed that there were fifty canonical equations of the form (3.1) with this property, now known as the Painlevé property. Further Painlevé, Gambier and their colleagues showed that of these fifty equations, forty-four can be reduced to linear equations, solved in terms of elliptic functions, or are reducible to one of six new nonlinear ordinary differential equations that define new transcendental functions, see Ince [46]. The Painlevé equations can be thought of as nonlinear analogues of the classical special functions [22,30,43,48,72], and arise in a wide variety of applications, for example random matrices, cf. [34,64].

Hamiltonian structure
Each of the Painlevé equations P I -P VI can be written as a Hamiltonian system for a suitable Hamiltonian function H J (q, p, z) [49,60,62]. The function σ(z) ≡ H J (q, p, z) satisfies a second-order, second-degree ordinary differential equation, whose solution is expressible in terms of the solution of the associated Painlevé equation [49,61,62]. The Hamiltonian associated with P IV (1.2) is with ϑ 0 and ϑ ∞ parameters [49,60,61,62], and so from (3.2) Solving (3.4a) for p and substituting in (3.4b) yields Analogously, solving (3.4b) for q and substituting in (3.4a) yields .
∞ . An important property of the Hamiltonian, which is very useful in applications, is that it satisfies a second-order, second-degree ordinary differential equation.
where q and p satisfy the system (3.4), then σ satisfies the second-order, second-degree ordinary differential equation Conversely, if σ is a solution of (3.5), then solutions of the Hamiltonian system (3.4) are given by Proof. See Jimbo and Miwa [49] and Okamoto [60,61,62].

Remarks 3.2.
1. Equation (3.5), which is often known as S IV (or the P IV σ-equation), is equivalent to equation SD-I.c in the classification of second order, second-degree ordinary differential equations with the Painlevé property by Cosgrove and Scoufis [24], an equation first derived and solved by Chazy [14] and subsequently by Bureau [12,13] by expressing the solution in terms of solutions of P IV .
2. Theorem 3.1 shows that solutions of equation (3.5) are in a one-to-one correspondence with solutions of the Hamiltonian system (3.4), and so are in a one-to-one correspondence with solutions of P IV (1.2).

Bäcklund and Schlesinger transformations
The Painlevé equations P II -P VI possess Bäcklund transformations which relate one solution to another solution either of the same equation, with different values of the parameters, or another equation (see [22,29,43] and the references therein). An important application of the Bäcklund transformations is that they generate hierarchies of classical solutions of the Painlevé equations, which are discussed in §3.3.
Bäcklund transformations for P IV (1.2) are given as follows. Then valid when the denominators are non-zero, and where the upper signs or the lower signs are taken throughout each transformation.
A class of Bäcklund transformations for the Painlevé equations is generated by so-called Schlesinger transformations of the associated isomonodromy problems. Fokas, Mugan and Ablowitz [33], deduced the following Schlesinger transformations R 1 -R 4 for P IV . 7c) where q ≡ q(z; A, B) and Fokas, Mugan and Ablowitz [33] also defined the composite transformations R 5 = R 1 R 3 and R 7 = R 2 R 4 given by We remark that R 5 and R 7 are the transformations T + and T − , respectively, given by Murata [58].

Special function solutions
The Painlevé equations P II -P VI possess hierarchies of solutions expressible in terms of classical special functions, for special values of the parameters through an associated Riccati equation, where f 2 (z), f 1 (z) and f 0 (z) are rational functions. Hierarchies of solutions, which are often referred to as "oneparameter solutions" (since they have one arbitrary constant), are generated from "seed solutions" derived from the Riccati equation using the Bäcklund transformations given in §3.2. Furthermore, as for the rational solutions, these special function solutions are often expressed in the form of determinants. Solutions of P II -P VI are expressed in terms of special functions as follows (see [22,43,56], and the references therein): for P II in terms of Airy functions Ai(z) and Bi(z); for P III in terms of Bessel functions J ν (z) and Y ν (z); for P IV in terms of parabolic cylinder functions functions D ν (z); for P V in terms of confluent hypergeometric functions 1 F 1 (a; c; z) (equivalently Kummer functions M (a, b, z) and U (a, b, z) or Whittaker functions M κ,µ (z) and W κ,µ (z)); and for P VI in terms of hypergeometric functions 2 F 1 (a, b; c; z). Some classical orthogonal polynomials arise as particular cases of these special function solutions and thus yield rational solutions of the associated Painlevé equations: for P III and P V in terms of associated Laguerre polynomials L (m) k (z); for P IV in terms of Hermite polynomials H n (z); and for P VI in terms of Jacobi polynomials P with n ∈ Z and ε = ±1.
For P IV (1.2) the associated Riccati equation is with P IV parameters A = −ε(ν + 1) and B = −2ν 2 . Letting w(z) = d dz ln ϕ ν (z) in (3.12) yields The solution of this equation depends on whether ν ∈ Z or ν ∈ Z, which we now summarize.
(i) If ν ∈ Z then equation (3.13) has solutions (3.14) with C 1 and C 2 arbitrary constants, where D ν (ζ) is the parabolic cylinder function which satisfies 15) and the boundary condition as ζ → +∞.
(ii) If ν = 0 then equation (3.13) has the solutions with C 1 and C 2 arbitrary constants, where erfc(z) is the complementary error function and erfi(z) is the imaginary error function, respectively defined by with C 1 and C 2 arbitrary constants.

Semi-classical Laguerre weight
In this section we consider monic orthogonal polynomials P n (x; t), for n ∈ N, with respect to the semi-classical Laguerre weight (1.1), where these polynomials satisfy the three-term recurrence relation (2.8), i.e. xP n (x; t) = P n+1 (x; t) + α n (t)P n (x; t) + β n (t)P n−1 (x; t),  x n−1 x n = y n + n + 1 2 λ y 2 n − 1 4 λ 2 , Boelen and van Assche [10] also show that the system (4.3) can be obtained from an asymmetric discrete P IV equation by a limiting process. However, from our point of view, it is more convenient to have the discrete system satisfied by α n and β n , which is given in the following Lemma.
Since the semi-classical Laguerre weight (1.1) has the form ω 0 (x) exp(xt) and the moments are finite for all t ∈ R, with t a parameter, then the coefficients α n (t) and β n (t) in the recurrence relation (4.1) satisfy the Toda system, recall Theorem 2.5.
We are now in a position to prove the relationship between the coefficients α n (t) and β n (t) in the recurrence relation (4.1) associated with the semi-classical Laguerre weight (1.1) and solutions of P IV (1.2). Theorem 4.3. The coefficients α n (t) and β n (t) in the recurrence relation (4.1) associated with the semi-classical Laguerre weight (1.1) are given by α n (t) = 1 2 q n (z) + 1 2 t, (4.5a) Proof. Solving the discrete system (4.4) for α n−1 and β n+1 yields and then substituting these into (2.11) gives Solving (4.8a) for β n yields and then substituting this into (4.8b) yields Making the transformation (4.5a) in this equation yields equation (4.6), which is P IV (1.2) with parameters given by (4.7). Finally making the transformation (4.5a) in (4.9) yields (4.5b), as required.
2. From Theorem 3.4 we see that the parameters (4.7) satisfy (3.10) with ε = −1, and therefore satisfy the condition given in Theorem 3.4 for P IV to have solutions expressible in terms of parabolic cylinder functions.
3. If q n is a solution of equation (4.6) then the solutions q n+1 and q n−1 are given by where ′ ≡ d/dz, which are special cases of the Schlesinger transformations R 5 (3.8a) and R 7 (3.8b), respectively.
Proof. The result is easily shown by comparing (4.11) and (3.14).
Having obtained an explicit expression for µ 0 we can now derive explicit expressions for the Hankel determinant ∆ n (t) and the coefficients α n (t) and β n (t) in the recurrence relation (4.1).
Proof. This is an immediate consequence of Theorem 2.1.
Proof. This is an immediate consequence of Theorems 2.1 and 2.3.
Remark 4.12. Differentiating (4.15) and letting S n (t) = d dt ln ∆ n (t) yields the fourth-order, bi-linear equation as is easily verified.
In Appendix 1 we give the first few recurrence coefficients for the semi-classical Laguerre weight (1.1) and the first few monic polynomials generated using the recurrence relation (4.1).
Since S n = d dt ln ∆ n then the asymptotic expansion (5.3) is easily derived from (5.2). Lemma 5.3. As t → ∞, the recurrence coefficients α n (t) and β n (t) have the asymptotic expansions Proof. By definition and so as t → ∞, as required. Consequently

Semi-classical Hermite weight
In this section we are concerned with the semi-classical Hermite weight ω(x; t) = |x| λ exp(−x 2 + tx), x, t ∈ R, λ > −1, which is an extension of the semi-classical Laguerre weight (1.1) to the whole real line, where we have ensured that the weight is positive by using |x| λ rather than x λ . Monic orthogonal polynomials associated with the semi-classical Hermite weight (6.1) satisfy the recurrence relation and our interest is in obtaining explicit expressions for the coefficients α n (t) and β n (t) in (6.2). First we evaluate the moment µ 0 (t; λ). Theorem 6.1. For the semi-classical Hermite weight (6.1), the moment µ 0 (t; λ) is given by with m ∈ N, where D ν (z) is the parabolic cylinder function, H n (z) is the Hermite polynomial and erf(z) is the error function.
Next we obtain an explicit expression for the Hankel determinant ∆ n (t).
Finally we obtain explicit expressions for the coefficients α n (t) and β n (t).