Semi-classical orthogonal polynomialsand the Painlevé equations

Peter A ClarksonSchool of Mathematics, Statistics and Actuarial Science

University of Kent, Canterbury, CT2 7NF, UKP.A.Clarkson@kent.ac.uk

Analytic, Algebraic and Geometric Aspects of Differential EquationsBedlewo, Poland

14 September 2015

Outline1. Introduction

Outline1. Introduction2. Properties of the second Painlevé equation

d2q

dz2 = 2q3 + zq + α

•Hamiltonian structure• Bäcklund transformations and associated difference equations• Airy solutions

Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation

d2q

dz2 =1

2q

(dq

dz

)2

+3

2q3 + 4zq2 + 2(z2 − α)q +

β

qPIV

•Hamiltonian structure• Parabolic cylinder function solutions

Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation4. Orthogonal polynomials

Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation4. Orthogonal polynomials5. Semi-classical orthogonal polynomials and the fourth Painlevé equation• Semi-classical Hermite weight

ω(x; t) = |x|ν exp(−x2 + tx), x, t ∈ R, ν > −1

•Generalized Freud weightω(x; t) = |x|2ν+1 exp

(−x4 + tx2), x, t ∈ R, ν > 0

Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation4. Orthogonal polynomials5. Semi-classical orthogonal polynomials and the fourth Painlevé equation6. Orthogonal polynomials on complex contours

ω(x; t) = exp(−1

3x3 + tx

), t > 0

on the curve C from e2πi/3∞ to e−2πi/3∞.

Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation4. Orthogonal polynomials5. Semi-classical orthogonal polynomials and the fourth Painlevé equation6. Orthogonal polynomials on complex contours7. Conclusions

Painlevé Equations

d2q

dz2 = 6q2 + z PI

d2q

dz2 = 2q3 + zq + A PII

d2q

dz2 =1

q

(dq

dz

)2

− 1

z

dq

dz+Aq2 + B

z+ Cq3 +

D

qPIII

d2q

dz2 =1

2q

(dq

dz

)2

+3

2q3 + 4zq2 + 2(z2 − A)q +

B

qPIV

d2q

dz2 =

(1

2q+

1

q − 1

)(dq

dz

)2

− 1

z

dq

dz+

(q − 1)2

z2

(Aq +

B

q

)PV

+Cq

z+Dq(q + 1)

q − 1

d2q

dz2 =1

2

(1

q+

1

q − 1+

1

q − z)(

dq

dz

)2

−(

1

z+

1

z − 1+

1

q − z)

dq

dzPVI

+q(q − 1)(q − z)

z2(z − 1)2

{A +

Bz

q2+C(z − 1)

(q − 1)2+Dz(z − 1)

(q − z)2

}with A, B, C and D arbitrary constants.

Painlevé σ-Equations(

d2σ

dz2

)2

+ 4

(dσ

dz

)3

+ 2zdσ

dz− 2σ = 0 SI(

d2σ

dz2

)2

+ 4

(dσ

dz

)3

+ 2dσ

dz

(z

dσ

dz− σ

)= 1

4β2 SII(

zd2σ

dz2 −dσ

dz

)2

+ 4

(dσ

dz

)2(z

dσ

dz− 2σ

)+ 4zϑ∞

dσ

dz= z2

(z

dσ

dz− 2σ + 2ϑ0

)SIII(

d2σ

dz2

)2

− 4

(z

dσ

dz− σ

)2

+ 4dσ

dz

(dσ

dz+ 2ϑ0

)(dσ

dz+ 2ϑ∞

)= 0 SIV(

zd2σ

dz2

)2

−[

2

(dσ

dz

)2

− zdσ

dz+ σ

]2

+ 4

4∏j=1

(dσ

dz+ κj

)= 0 SV

dσ

dz

[z(z − 1)

d2σ

dz2

]2

+

[dσ

dz

{2σ − (2z − 1)

dσ

dz

}+ κ1κ2κ3κ4

]2

=

4∏j=1

(dσ

dz+ κ2

j

)SVI

where β, ϑ0, ϑ∞ and κ1, . . . , κ4 are arbitrary constants.

Classical Special Functions• Airy, Bessel, Whittaker, Kummer, hypergeometric functions• Special solutions in terms of rational and elementary functions (for cer-

tain values of the parameters)• Solutions satisfy linear ordinary differential equations and linear dif-

ference equations• Solutions related by linear recurrence relations

Classical Special Functions• Airy, Bessel, Whittaker, Kummer, hypergeometric functions• Special solutions in terms of rational and elementary functions (for cer-

tain values of the parameters)• Solutions satisfy linear ordinary differential equations and linear dif-

ference equations• Solutions related by linear recurrence relations

Painlevé Transcendents — Nonlinear Special Functions• Special solutions such as rational solutions, algebraic solutions and spe-

cial function solutions (for certain values of the parameters)• Solutions satisfy nonlinear ordinary differential equations and nonlin-

ear difference equations• Solutions related by nonlinear recurrence relations

• The Painlevé equations are a chapter in the “Digital Library of Mathe-matical Functions", which is a rewrite/update of Abramowitz & Ste-gun’s “Handbook of Mathematical Functions". This was publishedonline, see http://dlmf.nist.gov/32, in 2010 and in the book “NISTHandbook of Mathematical Functions", by Cambridge UniversityPress [Edited by FWJ Olver, Lozier, Boisvert & Clark].

NotationSpecial Notation

PropertiesDifferential EquationsGraphicsIsomonodromy ProblemsIntegral EquationsHamiltonian StructureBäcklund TransformationsRational SolutionsOther Elementary SolutionsSpecial Function SolutionsAsymptotic Approximations for RealVariablesAsymptotic Approximations forComplex Variables

ApplicationsReductions of Partial DifferentialEquationsCombinatoricsOrthogonal PolynomialsPhysical

ComputationMethods of Computation

31.18 Methods of Computation 32.1 Special Notation

Chapter 32 Painlevé TranscendentsP. A. Clarkson

School of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, United Kingdom.

© 2010–2014 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.10; Release date 2015-08-07. A printed companion isavailable.

32.1

32.232.332.432.532.632.732.832.9

32.1032.11

32.12

32.13

32.1432.1532.16

32.17

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Annotate

Special function solutions of Painlevé equations

Number of(essential)

parameters

Specialfunction

Number ofparameters

Associatedorthogonalpolynomial

PI 0 —

PII 1Airy

Ai(z),Bi(z)0 —

PIII 2Bessel

Jν(z), Iν(z), Kν(z)1 —

PIV 2Parabolic

Dν(z)1

HermiteHn(z)

PV 3

KummerM(a, b, z), U(a, b, z)

WhittakerMκ,µ(z),Wκ,µ(z)

2

AssociatedLaguerreL

(k)n (z)

PVI 4hypergeometric

2F1(a, b; c; z)3

JacobiP (α,β)n (z)

Properties of the Second Painlevé Equation

d2q

dz2 = 2q3 + zq + α PII

• Hamiltonian structure• Bäcklund transformations and

associated difference equations• Airy solutions

Hamiltonian RepresentationPII can be written as the Hamiltonian system

dq

dz=∂HII

∂p= p− q2 − 1

2z,dp

dz= − ∂HII

∂q= 2qp + α + 1

2 HII

where HII(q, p, z;α) is the Hamiltonian defined by

HII(q, p, z;α) = 12p

2 − (q2 + 12z)p− (α + 1

2)q

Eliminating p then q satisfies PII whilst eliminating q yields

pd2p

dz2 =1

2

(dp

dz

)2

+ 2p3 − zp2 − 12(α + 1

2)2 P34

Theorem (Okamoto [1986])The function σ(z;α) = HII ≡ 1

2p2 − (q2 + 1

2z)p− (α + 12)q satisfies(

d2σ

dz2

)2

+ 4

(dσ

dz

)3

+ 2dσ

dz

(z

dσ

dz− σ

)= 1

4(α + 12)2 SII

and conversely

q(z;α) =2σ′′(z) + α + 1

2

4σ′(z), p(z;α) = −2

dσ

dz

is a solution of HII.

Bäcklund Transformations &Associated Difference Equations

Suppose that q(z;α) is a solution of PII

d2q

dz2 = 2q3 + zq + α

Then the Bäcklund transformations (Gambier [1910])

q(z;α + 1) = −q(z;α)− 2α + 1

2q2(z;α) + 2q′(z;α) + z

q(z;α− 1) = −q(z;α)− 2α− 1

2q2(z;α)− 2q′(z;α) + z

are also solutions of PII. Eliminating q′(z;α) yields2α + 1

q(z;α + 1) + q(z;α)+

2α− 1

q(z;α) + q(z;α− 1)+ 4q2(z;α) + 2z = 0

Hence settingqα±1 = q(z;α± 1), qα = q(z;α)

gives2α + 1

qα+1 + qα+

2α− 1

qα + qα−1+ 4q2

α + 2z = 0

which is known as alt-dPI (Fokas, Grammaticos & Ramani [1993]).

Airy Solutions of PII, P34 and SII

d2q

dz2 = 2q3 + zq + α PII

pd2p

dz2 =1

2

(dp

dz

)2

+ 2p3 − zp2 − 12(α + 1

2)2 P34(d2σ

dz2

)2

+ 4

(dσ

dz

)3

+ 2dσ

dz

(z

dσ

dz− σ

)= 1

4(α + 12)2 SII

TheoremPII, P34 and SII have solutions expressible in terms of the Riccati equation

εdq

dz= q2 + 1

2z, ε = ±1 (1)

if and only if α = n+ 12, with n ∈ Z, which has solution

q(z) = −ε d

dzlnϕ(z)

whereϕ(z) = cos(1

2θ) Ai(ζ) + sin(12θ) Bi(ζ), ζ = −2−1/2z

with Ai(ζ) and Bi(ζ) the Airy functions and θ is an arbitrary constant.

Airy Solutions of PII, P34 and SII

d2qn

dz2 = 2q3n + zqn + n + 1

2 PII

pnd2pn

dz2 =1

2

(dpndz

)2

+ 2p3n − zp2

n − 12n

2 P34(d2σn

dz2

)2

+ 4

(dσndz

)3

+ 2dσndz

(z

dσndz− σ

)= 1

4n2 SII

TheoremLet

ϕ(z; θ) = cos(12θ) Ai(ζ) + sin(1

2θ) Bi(ζ), ζ = −2−1/2z

with θ an arbitrary constant, Ai(ζ) and Bi(ζ) Airy functions, and τn(z) bethe Wronskian

τn(z; θ) =W(ϕ,

dϕ

dz, . . . ,

dn−1ϕ

dzn−1

)then

qn(z; θ) =d

dzln

τn(z; θ)

τn+1(z; θ), pn(z; θ) =

d2

dz2 ln τn(z; θ), σn(z; θ) =d

dzln τn(z; θ)

respectively satisfy PII, P34 and SII, with n ∈ Z.

Airy Solutions of PII qn(z; θ) =d

dzln

τn(z; θ)

τn+1(z; θ)

n = 0, θ = 0, 13π,

23π, π n = 1, θ = 0, 1

3π,23π, π

n = 2, θ = 0, 13π,

23π, π n = 3, θ = 0, 1

3π,23π, π

Airy Solutions of PII with α = 52

(Fornberg and Weideman [2014])

q(z; 52) =

d

dzlnW(ϕ, ϕ′)W(ϕ, ϕ′, ϕ′′)

, ϕ(z) = cos(12θ) Ai(−2−1/3z) + sin(1

2θ) Bi(−2−1/3z)

blue/yellow denote poles with residue +1/− 1

Airy Solutions of P34 pn(z; θ) =d2

dz2ln τn(z; θ)

n = 1, θ = 0, 13π,

23π, π n = 2, θ = 0, 1

3π,23π, π

n = 3, θ = 0, 13π,

23π, π n = 4, θ = 0, 1

3π,23π, π

Airy Solutions of P34 pn(z; θ) =d2

dz2ln τn(z; θ)

Plots of p(z; 0)/n for n = 2,4,6,8

Airy Solutions of SII σn(z; θ) =d

dzln τn(z; θ)

n = 1, θ = 0, 13π,

23π, π n = 2, θ = 0, 1

3π,23π, π

n = 3, θ = 0, 13π,

23π, π n = 4, θ = 0, 1

3π,23π, π

Airy Solutions of SII σn(z; 0) =d

dzlnW

(ϕ, ϕ′, . . . , ϕ(n−1)

), ϕ = Ai(−2−1/3z)

Plots of σn(z; 0)/n for n = 2,4,6,8

Properties of the Fourth Painlevé Equationand the Fourth Painlevé σ-Equation

d2q

dz2 =1

2q

(dq

dz

)2

+3

2q3 + 4zq2 + 2(z2 − A)q +

B

qPIV

(d2σ

dz2

)2

− 4

(z

dσ

dz− σ

)2

+ 4dσ

dz

(dσ

dz+ 2ϑ0

)(dσ

dz+ 2ϑ∞

)= 0 SIV

• Hamiltonian Representation• Parabolic Cylinder Function Solutions

Hamiltonian Representation of PIVPIV can be written as the Hamiltonian system

dq

dz=∂HIV

∂p= 4qp− q2 − 2zq − 2ϑ0

dp

dz= − ∂HIV

∂q= −2p2 + 2pq + 2zp− ϑ∞

where HIV(q, p, z;ϑ0, ϑ∞) is the Hamiltonian defined by

HIV(q, p, z;ϑ0, ϑ∞) = 2qp2 − (q2 + 2zq + 2ϑ0)p + ϑ∞q

Eliminating p then q satisfies

d2q

dz2 =1

2q

(dq

dz

)2

+ 32q

3 + 4zq2 + 2(z2 + ϑ0 − 2ϑ∞ − 1)q − 2ϑ20

q

which is PIV with A = 1 − ϑ0 + 2ϑ∞ and B = −2ϑ20, whilst eliminating q then

p satisfies

d2p

dz2 =1

2p

(dq

dz

)2

+ 6p3 − 8zp2 + 2(z2 − 2ϑ0 + ϑ∞ + 1)p− ϑ2∞

2p

and letting p = −12q gives PIV with A = 2ϑ0 − ϑ∞ − 1 and B = −2ϑ2

∞.

Theorem (Okamoto [1986])The function

σ(z;ϑ0, ϑ∞) = HIV ≡ 2qp2 − (q2 + 2zq + 2ϑ0)p + ϑ∞q

where q and p satisfy the Hamiltonian systemdq

dz= 4qp− q2 − 2zq − 2ϑ0,

dp

dz= −2p2 + 2pq + 2zp− ϑ∞ HIV

satisfies the second-order, second-degree equation(d2σ

dz2

)2

− 4

(z

dσ

dz− σ

)2

+ 4dσ

dz

(dσ

dz+ 2ϑ0

)(dσ

dz+ 2ϑ∞

)= 0 SIV

Conversely, if σ(z;ϑ0, ϑ∞) is a solution of SIV, then

q(z;ϑ0, ϑ∞) =σ′′ − 2zσ′ + 2σ

2(σ′ + 2ϑ∞), p(z;ϑ0, ϑ∞) =

σ′′ + 2zσ′ − 2σ

4(σ′ + 2ϑ0)

are solutions of the Hamiltonian system HIV.

Parabolic Cylinder Function Solutions of PIVTheorem

Suppose τν,n(z; ε) is given by

τν,n(z; ε) =W(ϕν(z; ε), ϕ′ν(z; ε), . . . , ϕ(n−1)

ν (z; ε)), n ≥ 1

where τν,0(z; ε) = 1 and ϕν(z; ε) satisfiesd2ϕν

dz2 − 2εzdϕνdz

+ 2ενϕν = 0, ε2 = 1

Then solutions of PIV

d2q

dz2 =1

2q

(dq

dz

)2

+3

2q3 + 4zq2 + 2(z2 − A)q +

B

q

are given by

q[1]ν,n(z) = −2z + ε

d

dzlnτν,n+1(z; ε)

τν,n(z; ε), (A1, B1) =

(ε(2n− ν),−2(ν + 1)2

)q[2]ν,n(z) = ε

d

dzlnτν,n+1(z; ε)

τν+1,n(z; ε), (A2, B2) =

(− ε(n + ν),−2(ν − n + 1)2)

q[3]ν,n(z) = −ε d

dzlnτν+1,n(z; ε)

τν,n(z; ε), (A3, B3) =

(ε(2ν − n + 1),−2n2

)

Parabolic Cylinder Function Solutions of SIV

TheoremSuppose τν,n(z; ε) is given by

τν,n(z; ε) =W(ϕν(z; ε), ϕ′ν(z; ε), . . . , ϕ(n−1)

ν (z; ε)), n ≥ 1

where τν,0(z; ε) = 1 and ϕν(z; ε) satisfies

d2ϕν

dz2 − 2εzdϕνdz

+ 2ενϕν = 0, ε2 = 1

Then solutions of SIV(d2σ

dz2

)2

− 4

(z

dσ

dz− σ

)2

+ 4dσ

dz

(dσ

dz+ 2ϑ0

)(dσ

dz+ 2ϑ∞

)= 0

are given by

σν,n(z) =d

dzln τν,n(z; ε), (ϑ0, ϑ∞) =

(ε(ν − n + 1),−εn)

d2ϕν

dz2 − 2εzdϕνdz

+ 2ενϕν = 0, ε2 = 1 (∗)

• If ν 6∈ Z

ϕν(z; ε) =

{{C1Dν(

√2 z) + C2Dν(−

√2 z)}

exp(

12z

2), if ε = 1{

C1D−ν−1(√

2 z) + C2D−ν−1(−√

2 z)}

exp(−1

2z2), if ε = −1

• If ν = n ∈ Z, with n ≥ 0

ϕn(z; ε) =

C1Hn(z) + C2 exp(z2)

dn

dzn{

erfi(z) exp(−z2)}, if ε = 1

C1Hn(iz) + C2 exp(−z2)dn

dzn{

erfc(z) exp(z2)}, if ε = −1

• If ν = −n ∈ Z, with n ≥ 1

ϕ−n(z; ε) =

C1Hn−1(iz) exp(z2) + C2

dn−1

dzn−1

{erfc(z) exp(z2)

}, if ε = 1

C1Hn−1(z) exp(−z2) + C2dn−1

dzn−1

{erfi(z) exp(−z2)

}, if ε = −1

with C1 and C2 arbitrary constants, Dν(ζ) the parabolic cylinder func-tion, Hn(z) the Hermite polynomial, erfc(z) the complementary errorfunction and erfi(z) the imaginary error function.

Orthogonal Polynomials

• Some History

• Monic orthogonal polynomials

• Semi-classical orthogonal polynomials

Some History• The relationship between semi-classical orthogonal polynomials and in-

tegrable equations dates back to Shohat [1939], Freud [1976], Bonan& Nevai [1984].• Fokas, Its & Kitaev [1991, 1992] identified these integrable equations

as discrete Painlevé equations.•Magnus [1995] considered the Freud weight

ω(x; t) = exp(−x4 + tx2

), x, t ∈ R,

and showed that the coefficients in the three-term recurrence relation canbe expressed in terms of solutions of

qn(qn−1 + qn + qn+1) + 2tqn = n

which is discrete PI (dPI), as shown by Bonan & Nevai [1984], and

d2qn

dt2=

1

2qn

(dqndt

)2

+3

2q3n + 4tq2

n + 2(t2 + 12n)qn − n2

2qn

which is PIV with A = −12n and B = −1

2n2. The connection between the

Freud weight and solutions of dPI and PIV is due to Kitaev [1988].

Extract from Digital Library of Mathematical Functions18.32 OP’s with Respect to Freud Weights 475

18.32 OP’s with Respect to Freud Weights

A Freud weight is a weight function of the form

18.32.1 w(x) = exp(−Q(x)), −∞ < x <∞,

where Q(x) is real, even, nonnegative, and continu-ously differentiable. Of special interest are the casesQ(x) = x2m, m = 1, 2, . . . . No explicit expressionsfor the corresponding OP’s are available. However, forasymptotic approximations in terms of elementary func-tions for the OP’s, and also for their largest zeros, seeLevin and Lubinsky (2001) and Nevai (1986). For auniform asymptotic expansion in terms of Airy func-tions (§9.2) for the OP’s in the case Q(x) = x4 see Boand Wong (1999).

18.33 Polynomials Orthogonal on the UnitCircle

18.33(i) Definition

A system of polynomials {φn(z)}, n = 0, 1, . . . , whereφn(z) is of proper degree n, is orthonormal on the unitcircle with respect to the weight function w(z) (≥ 0) if

18.33.11

2πi

∫|z| =1

φn(z)φm(z)w(z)dz

z= δn,m,

where the bar signifies complex conjugate. See Simon(2005a,b) for general theory.

18.33(ii) Recurrence Relations

Denote

18.33.2 φn(z) = κnzn +

n∑`=1

κn,n−`zn−`,

where κn(> 0), and κn,n−`(∈ C) are constants. Alsodenote

18.33.3 φ∗n(z) = κnzn +

n∑`=1

κn,n−`zn−`,

where the bar again signifies compex conjugate. Then

18.33.4 κnzφn(z) = κn+1φn+1(z)− φn+1(0)φ∗n+1(z),

18.33.5 κnφn+1(z) = κn+1zφn(z) + φn+1(0)φ∗n(z),

18.33.6κnφn(0)φn+1(z) + κn−1φn+1(0)zφn−1(z)

= (κnφn+1(0) + κn+1φn(0)z)φn(z).

18.33(iii) Connection with OP’s on the Line

Assume that w(eiφ) = w(e−iφ). Set

18.33.7

w1(x) = (1− x2)−12w(x+ i(1− x2)

12

),

w2(x) = (1− x2)12w(x+ i(1− x2)

12

).

Let {pn(x)} and {qn(x)}, n = 0, 1, . . . , be OP’s withweight functions w1(x) and w2(x), respectively, on(−1, 1). Then18.33.8

pn(

12 (z + z−1)

)= (const.)× (z−nφ2n(z) + znφ2n(z−1)

)= (const.)× (z−n+1φ2n−1(z) + zn−1φ2n−1(z−1)

),

18.33.9

qn(

12 (z + z−1)

)= (const.)× z−n−1φ2n+2(z)− zn+1φ2n+2(z−1)

z − z−1

= (const.)× z−nφ2n+1(z)− znφ2n+1(z−1)z − z−1

.

Conversely,18.33.10

z−nφ2n(z)= Anpn

(12 (z + z−1)

)+Bn(z − z−1)qn−1

(12 (z + z−1)

),

18.33.11

z−n+1φ2n−1(z)= Cnpn

(12 (z + z−1)

)+Dn(z − z−1)qn−1

(12 (z + z−1)

),

where An, Bn, Cn, and Dn are independent of z.

18.33(iv) Special Cases

Trivial

18.33.12 φn(z) = zn, w(z) = 1.Szego–Askey

18.33.13

φn(z)

=n∑`=0

(λ+ 1)`(λ)n−``! (n− `)! z` =

(λ)nn! 2F1

( −n, λ+ 1−λ− n+ 1

; z),

with

18.33.14

w(z) =(1− 1

2 (z + z−1))λ,

w1(x) = (1− x)λ−12 (1 + x)−

12 ,

w2(x) = (1− x)λ+ 12 (1 + x)

12 , λ > − 1

2 .

For the hypergeometric function 2F1 see §§15.1 and15.2(i).Askey

18.33.15

φn(z) =n∑`=0

(aq2; q2

)`

(a; q2

)n−`

(q2; q2)` (q2; q2)n−`(q−1z)`

=

(a; q2

)n

(q2; q2)n2φ1

(aq2, q−2n

a−1q2−2n; q2,

qz

a

),

with18.33.16 w(z) =

∣∣∣(qz; q2)∞/ (

aqz; q2)∞

∣∣∣2 , a2q2 < 1.

For the notation, including the basic hypergeometricfunction 2φ1, see §§17.2 and 17.4(i).

When a = 0 the Askey case is also known as theRogers–Szego case.

Monic Orthogonal PolynomialsLet Pn(x), n = 0, 1, 2, . . . , be the monic orthogonal polynomials of degreen in x, with respect to the positive weight ω(x), such that∫ b

a

Pm(x)Pn(x)ω(x) dx = hnδm,n, hn > 0, m, n = 0, 1, 2, . . .

One of the important properties that orthogonal polynomials have is thatthey satisfy the three-term recurrence relation

xPn(x) = Pn+1(x) + αnPn(x) + βnPn−1(x)

where the recurrence coefficients are given by

αn =∆n+1

∆n+1− ∆n

∆n, βn =

∆n+1∆n−1

∆2n

with

∆n =

∣∣∣∣∣∣∣∣µ0 µ1 . . . µn−1

µ1 µ2 . . . µn... ... . . . ...

µn−1 µn . . . µ2n−2

∣∣∣∣∣∣∣∣ , ∆n =

∣∣∣∣∣∣∣∣µ0 µ1 . . . µn−2 µnµ1 µ2 . . . µn−1 µn+1... ... . . . ... ...

µn−1 µn . . . µ2n−3 µ2n−1

∣∣∣∣∣∣∣∣and µk =

∫ b

a

xk ω(x) dx are the moments of the weight ω(x).

Further Properties• The Hankel determinant

∆n =

∣∣∣∣∣∣∣∣µ0 µ1 . . . µn−1

µ1 µ2 . . . µn... ... . . . ...

µn−1 µn . . . µ2n−2

∣∣∣∣∣∣∣∣ , µk =

∫ b

a

xk ω(x) dx

also has the integral representation

∆n =1

n!

∫ b

a

· · ·n∫ b

a

n∏`=1

ω(x`)∏

1≤j<k≤n(xj − xk)2 dx1 . . . dxn, n ≥ 1

which is the partition function in random matrix theory.• The monic polynomials Pn(x) can be uniquely expressed as

Pn(x) =1

∆n

∣∣∣∣∣∣∣∣∣∣µ0 µ1 . . . µnµ1 µ2 . . . µn+1... ... . . . ...

µn−1 µn . . . µ2n−1

1 x . . . xn

∣∣∣∣∣∣∣∣∣∣• The normalization constants can be expressed as

hn =∆n+1

∆n, h0 = ∆1 = µ0

Example — Hermite polynomialsHermite polynomials are orthogonal with respect to the weight

ω(x) = exp(−x2), x ∈ RIn this case

µ2k =

∫ ∞−∞

x2k exp(−x2) dx =

√π (2k)!

22k k!, µ2k+1 =

∫ ∞−∞

x2k+1 exp(−x2) dx = 0

so

∆n =

∣∣∣∣∣∣∣∣µ0 µ1 . . . µn−1

µ1 µ2 . . . µn... ... . . . ...

µn−1 µn . . . µ2n−2

∣∣∣∣∣∣∣∣ = (12)n(n−1)/2

n−1∏k=1

(k!), ∆n = 0

and therefore

αn = 0, βn =∆n+1∆n−1

∆2n

= 12n

which gives the three-term recurrence relationPn+1(x) = xPn(x)− 1

2nPn−1(x)

wherePn(x) = 2−nHn(x)

with Hn(x) the Hermite polynomial.

Semi-classical Orthogonal PolynomialsConsider the Pearson equation satisfied by the weight ω(x)

d

dx[σ(x)ω(x)] = τ (x)ω(x)

• Classical orthogonal polynomials: σ(x) and τ (x) are polynomials withdeg(σ) ≤ 2 and deg(τ ) = 1

ω(x) σ(x) τ (x)

Hermite exp(−x2) 1 −2x

Laguerre xν exp(−x) x 1 + ν − xJacobi (1− x)α(1 + x)β 1− x2 β − α− (2 + α + β)x

• Semi-classical orthogonal polynomials: σ(x) and τ (x) are polynomi-als with either deg(σ) > 2 or deg(τ ) > 1

ω(x) σ(x) τ (x)

Airy exp(−13x

3 + tx) 1 t− x2

semi-classical Hermite |x|ν exp(−x2 + tx) x 1 + ν + tx− 2x2

Generalized Freud |x|2ν+1 exp(−x4 + tx2) x 2ν + 2 + 2tx2 − 4x4

If the weight has the form

ω(x; t) = ω0(x) exp(tx)

where the integrals∫ ∞−∞

xkω0(x) exp(tx) dx exist for all k ≥ 0.

• The recurrence coefficients αn(t) and βn(t) satisfy the Toda systemdαndt

= βn − βn+1,dβndt

= βn(αn − αn−1)

• The kth moment is given by

µk(t) =

∫ ∞−∞

xkω0(x) exp(tx) dx =dk

dtk

(∫ ∞−∞

ω0(x) exp(tx) dx

)=

dkµ0

dtk

• Since µk(t) =dkµ0

dtk, then ∆n(t) and ∆n(t) can be expressed as Wronskians

∆n(t) =W(µ0,

dµ0

dt, . . . ,

dn−1µ0

dtn−1

)= det

[dj+kµ0

dtj+k

]n−1

j,k=0

∆n(t) =W(µ0,

dµ0

dt, . . . ,

dn−2µ0

dtn−2 ,dnµ0

dtn

)=

d

dt∆n(t)

Semi-classical Hermite Weight

ω(x; t) = |x|ν exp(−x2 + tx), x ∈ R, ν > −1

• PAC & K Jordaan, “The relationship between semi-classical Laguerrepolynomials and the fourth Painlevé equation", Constr. Approx., 39 (2014)223–254

Semi-classical Hermite weightConsider the semi-classical Hermite weight

ω(x; t) = |x|ν exp(−x2 + tx), x ∈ R, ν > −1

• If ν 6∈ N, then the moment µ0(t; ν) is given by

µ0(t; ν) =

∫ ∞−∞|x|ν exp(−x2 + tx) dx

=

∫ ∞0

xν exp(−x2 + tx) dx +

∫ ∞0

xν exp(−x2 − tx) dx

=Γ(ν + 1) exp(1

8t2)

2(ν+1)/2

{D−ν−1

(− 12

√2 t)

+ D−ν−1

(12

√2 t)}

since the parabolic cylinder functionDν(ζ) has the integral representation

Dν(ζ) =exp(−1

4ζ2)

Γ(−ν)

∫ ∞0

s−ν−1 exp(−12s

2 − ζs) ds

• If ν = 2N , with N ∈ N then

µ0(t; 2N) =

∫ ∞−∞

x2N exp(−x2 + tx) dx =√π(− 1

2i)2N

H2N

(12it)

exp(

14t

2)

since the Hermite polynomial, Hn(z), has the integral representation

Hn(z) =2n√π

∫ ∞−∞

(z + ix)n exp(−x2) dx

• If ν = 2N + 1, with N ∈ N then

µ0(t; 2N + 1) =

∫ ∞−∞

x2N |x| exp(−x2 + tx) dx =√π

d2N+1

dt2N+1

{erf(1

2t) exp(

14t

2)}

as for n ∈ N,∫ ∞−∞

xn|x| exp(−x2 + tx) dx

=dn

dtn

{∫ ∞0

x exp(−x2 + tx) dx +

∫ ∞0

x exp(−x2 − tx) dx

}=

dn+1

dtn+1

{∫ ∞0

exp(−x2 + tx) dx−∫ ∞

0

exp(−x2 − tx) dx

}=

dn+1

dtn+1

[12

√π{

1 + erf(12t)]

exp(

14t

2)− 1

2

√π[1− erf(1

2t)]

exp(

14t

2)}

=√π

dn+1

dtn+1

{erf(1

2t) exp(

14t

2)}

since ∫ ∞0

exp(−x2 + tx) dx = 12

√π{

1 + erf(12t)}

exp(

14t

2)

• The moment µk(t; ν) is given by

µk(t; ν) =

∫ ∞−∞

xk|x|ν exp(−x2 + tx) dx

=dk

dtk

(∫ ∞−∞|x|ν exp(−x2 + tx) dx

)=

dkµ0

dtk

• The Hankel determinant ∆n(t; ν) is given by

∆n(t; ν) = det[µj+k(t; ν)

]n−1

j,k=0≡ W

(µ0,

dµ0

dt, . . . ,

dn−1µ0

dtn−1

)where

µ0(t; ν) =

Γ(ν + 1) exp(18t

2)

2(ν+1)/2

{D−ν−1

(− 12

√2 t)

+ D−ν−1

(12

√2 t)}, ν 6∈ N

√π(− 1

2i)2N

H2N

(12it)

exp(

14t

2), ν = 2N

√π

d2N+1

dt2N+1

{erf(1

2t) exp(

14t

2)}, ν = 2N + 1

Theorem (PAC & Jordaan [2014])The recurrence coefficients αn(t; ν) and βn(t; ν) in the three-term recurrence

relationxPn(x; t) = Pn+1(x; t) + αn(t; ν)Pn(x; t) + βn(t; ν)Pn−1(x; t),

for monic polynomials orthogonal with respect to the semi-classical Hermiteweight

ω(x; t) = |x|ν exp(−x2 + tx), x ∈ R, ν > −1

are given by

αn(t; ν) =d

dtln

∆n+1(t; ν)

∆n(t; ν), βn(t; ν) =

d2

dt2ln ∆n(t; ν)

where ∆n(t; ν) is the Hankel determinant

∆n(t; ν) = det[µj+k(t; ν)

]n−1

j,k=0≡ W

(µ0,

dµ0

dt, . . . ,

dn−1µ0

dtn−1

)with

µ0(t; ν) =

Γ(ν + 1) exp(18t

2)

2(ν+1)/2

{D−ν−1

(− 12

√2 t)

+ D−ν−1

(12

√2 t)}, ν 6∈ N

√π(− 1

2i)2N

H2N

(12it)

exp(

14t

2), ν = 2N

√π

d2N+1

dt2N+1

{erf(1

2t) exp(

14t

2)}, ν = 2N + 1

Remarks:• The Hankel determinant ∆n(t; ν) satisfies the Toda equation

d2

dt2ln ∆n(t; ν) =

∆n−1(t; ν)∆n+1(t; ν)

∆2n(t; ν)

and the fourth-order, bi-linear equation

∆nd4∆n

dt4− 4

d3∆n

dt3d∆n

dt+ 3

(d2∆n

dt2

)2

− (14t

2 + 4n + 2ν)

{∆n

d2∆n

dt2−(

d∆n

dt

)2}

+ 14t∆n

d∆n

dt+ 1

2n(n + ν)∆2n = 0

• The function Sn(t; ν) =d

dtln ∆n(t; ν) satisfies

4

(d2Sn

dt2

)2

−(tdSndt− Sn

)2

+ 4dSndt

(2

dSndt− n

)(2

dSndt− n− ν

)= 0

which is equivalent to SIV, the PIV σ-equation (let Sn(t; ν) = 12σ(z), with

z = 2t), so

αn(t; ν) = Sn+1(t; ν)− Sn(t; ν), βn(t; ν) =d

dtSn(t; ν)

Recurrence coefficients for ω(x; t) = x2 exp(−x2 + tx)

α0(t) = 12t +

2t

t2 + 2

α1(t) = 12t +

4t3

t4 + 12− 2t

t2 + 2

α2(t) = 12t +

6t(t4 + 12− 4t2)

t6 − 6t4 + 36t2 + 72− 4t3

t4 + 12

α3(t) = 12t +

8t3(t4 + 60− 12t2)

t8 − 16t6 + 120t4 + 720− 6t(t4 + 12− 4t2)

t6 − 6t4 + 36t2 + 72

α4(t) = 12t +

10t(t8 + 216t4 + 720− 24t6 − 480t2)

t10 − 30t8 + 360t6 − 1200t4 + 3600t2 + 7200− 8t3(t4 + 60− 12t2)

t8 − 16t6 + 120t4 + 720

β1(t) = 12 −

2(t2 − 2)

(t2 + 2)2

β2(t) = 1− 4t2(t2 − 6)(t2 + 6)

(t4 + 12)2

β3(t) = 32 −

6(t4 − 12t2 + 12)(t6 + 6t4 + 36t2 − 72)

(t6 − 6t4 + 36t2 + 72)2

β4(t) = 2− 8t2(t4 − 20t2 + 60)(t8 + 72t4 − 2160)

(t8 − 16t6 + 120t4 + 720)2

Hence, using the three-term recurrence relationPn+1(x; t) = [x− αn(t)]Pn(x; t)− βn(t)Pn−1(x; t), n = 0, 1, 2, . . .

then

P1(x; t) = x− t(t2 + 6)

2(t2 + 2)

P2(x; t) = x2 − t(t4 + 4t2 + 12)

t4 + 12x +

t6 + 6t4 + 36t2 − 72

4(t4 + 12)

P3(x; t) = x3 − 3t(t6 − 2t4 + 20t2 + 120)

2(t6 − 6t4 + 36t2 + 72)x2 +

3(t8 + 40t4 − 240)

4(t6 − 6t4 + 36t2 + 72)x

− t(t8 + 72t4 − 2160)

8(t6 − 6t4 + 36t2 + 72)

P4(x; t) = x4 − 2t(t8 − 12t6 + 72t4 + 240t2 + 720)

t8 − 16t6 + 120t4 + 720x3

+3(t10 − 10t8 + 80t6 + 1200t2 − 2400)

2(t8 − 16t6 + 120t4 + 720)x2

− t(t10 − 10t8 + 120t6 − 240t4 − 1200t2 − 7200)

2(t8 − 16t6 + 120t4 + 720)x

+t12 − 12t10 + 180t8 − 480t6 − 3600t4 − 43200t2 + 43200

16(t8 − 16t6 + 120t4 + 720)

Generalized Freud Weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

), x ∈ R, ν > 0

• PAC, K Jordaan, & A Kelil, “On a generalized Freud weight", preprint(2015).

Generalized Freud weightFor the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

), x ∈ R

the moments are

µ0(t; ν) =

∫ ∞−∞|x|2ν+1 exp

(−x4 + tx2)

dx

=

∫ ∞0

yν+1 exp(−y2 + ty

)dy

= 2−(ν+1)/2Γ(ν + 1) exp(18t

2)D−ν−1

(− 12

√2 t)

µ2n(t; ν) =

∫ ∞−∞

x2n|x|2ν+1 exp(−x4 + tx2

)dx

= (−1)ndn

dtn

(∫ ∞−∞|x|2ν+1 exp

(−x4 + tx2)

dx

)= (−1)n

dnµ0

dtn, n = 1, 2, . . .

µ2n+1(t; ν) =

∫ ∞−∞

x2n+1|x|2ν+1 exp(−x4 + tx2

)dx

= 0, n = 1, 2, . . .

We note that

µ2n(t; ν) =

∫ ∞−∞

x2n |x|2ν+1 exp(−x4 + tx2

)dx

=

∫ ∞−∞|x|2ν+2n+1 exp

(−x4 + tx2)dx

= µ0(t; ν + n).

Also, when ν = n ∈ Z+, then

D−n−1

(− 12

√2 t)

= 12

√2π

dn

dtn{[

1 + erf(

12t)]

exp(

18t

2)},

where erf(z) is the error function.

Theorem (PAC, Jordaan & Kelil [2015])The recurrence coefficient βn(t) in the three-term recurrence relation

xPn(x; t) = Pn+1(x; t) + βn(t)Pn−1(x; t),

is given by

β2n(t; ν) =d

dtlnτn(t; ν + 1)

τn(t; ν), β2n+1(t; ν) =

d

dtln

τn+1(t; ν)

τn(t; ν + 1)

where τn(t; ν) is the Wronskian given by

τn(t; ν) =W(φν,

dφνdt, . . . ,

dn−1φν

dtn−1

)with

φν(t) = µ0(t; ν) =Γ(ν + 1)

2(ν+1)/2exp(

18t

2)D−ν−1

(− 12

√2 t)

Theorem (PAC, Jordaan & Kelil [2015])The recurrence coefficient βn(t) in the three-term recurrence relation

xPn(x; t) = Pn+1(x; t) + βn(t)Pn−1(x; t),

is given by

β2n(t; ν) =d

dtlnτn(t; ν + 1)

τn(t; ν), β2n+1(t; ν) =

d

dtln

τn+1(t; ν)

τn(t; ν + 1)

where τn(t; ν) is the Wronskian given by

τn(t; ν) =W(φν,

dφνdt, . . . ,

dn−1φν

dtn−1

)with

φν(t) = µ0(t; ν) =Γ(ν + 1)

2(ν+1)/2exp(

18t

2)D−ν−1

(− 12

√2 t)

Remark: The function Sn(t; ν) =d

dtln τn(t; ν) satisfies

4

(d2Sn

dt2

)2

−(tdSndt− Sn

)2

+ 4dSndt

(2

dSndt− n

)(2

dSndt− n− ν

)= 0

which is equivalent to SIV, the PIV σ-equation, soβ2n(t; ν) = Sn(t; ν + 1)− Sn(t; ν), β2n+1(t; ν) = Sn(t + 1; ν)− Sn(t; ν + 1)

TheoremThe recurrence coefficients βn(t) satisfy the equation

d2βn

dt2=

1

2βn

(dβndt

)2

+ 32β

3n − tβ2

n + (18t

2 − 12An)βn +

Bn

16βn(1)

which is equivavlent to PIV, where the parameters An and Bn are given byA2n = −2ν − n− 1, A2n+1 = ν − nB2n = −2n2, B2n+1 = −2(ν + n + 1)2

Further βn(t) satisfies the nonlinear difference equation

βn+1 + βn + βn−1 = 12t +

2n + (2ν + 1)[1− (−1)n]

8βn(2)

which is known as discrete PI.

TheoremThe recurrence coefficients βn(t) satisfy the equation

d2βn

dt2=

1

2βn

(dβndt

)2

+ 32β

3n − tβ2

n + (18t

2 − 12An)βn +

Bn

16βn(1)

which is equivavlent to PIV, where the parameters An and Bn are given byA2n = −2ν − n− 1, A2n+1 = ν − nB2n = −2n2, B2n+1 = −2(ν + n + 1)2

Further βn(t) satisfies the nonlinear difference equation

βn+1 + βn + βn−1 = 12t +

2n + (2ν + 1)[1− (−1)n]

8βn(2)

which is known as discrete PI.

Remark: The link between the differential equation (1) and the differenceequation (2) is given by the Bäcklund transformations

βn+1 =1

2βn

dβndt− 1

2βn + 14t +

cn4βn

, βn−1 = − 1

2βn

dβndt− 1

2βn + 14t +

cn4βn

with cn = 12n + 1

4(2ν + 1)[1− (−1)n].

The first few recurrence coefficients are:β1(t) = Φν

β2(t) = −2Φ2ν − tΦν − ν − 1

2Φν

β3(t) = − Φν

2Φ2ν − tΦν − ν − 1

− ν + 1

2Φν

β4(t) =t

2(ν + 2)+

Φν

2Φ2ν − tΦν − ν − 1

+(ν + 1)(t2 + 2ν + 4)Φν + (ν + 1)2t

2(ν + 2)[2(ν + 2)Φ2ν − (ν + 1)tΦν − (ν + 1)2]

β5(t) = − 2νt

ν + 1− 2(ν + 1)

t− 2ν(2t2 + ν + 1)Φν − 4ν2t

(ν + 1)[(ν + 1)Φ2ν + 2νtΦν − 2ν2]

− 2[νt2 + (ν + 1)(2ν + 1)]Φ2ν + 2νt(t2 + 4ν + 5)Φν − 4ν2t2 − 8ν2(ν + 1)

t[tΦ3ν(t) + (2t2 − 2ν + 1)Φ2

ν − 6Φννt + 4ν2]

where

Φν(t) =d

dtln{D−ν−1

(− 12

√2 t)

exp(

18t

2)}

= 12t + 1

2

√2D−ν

(− 12

√2 t)

D−ν−1

(− 12

√2 t).

Hence, using the three-term recurrence relation

Pn+1(x; t) = xPn(x; t)− βn(t)Pn−1(x; t), n = 0, 1, 2, . . .

withP0(x; t) = 1, P−1(x; t) = 0, β0(t) = 0

then the first few polynomials are given byP1(x; t) = x

P2(x; t) = x2 − Φν

P3(x; t) = x3 + 2tΦν − ν

Φνx

P4(x; t) = x4 + 2tΦ2

ν + (2t2 + 1)Φν − 2νt

Φ2ν + 2tΦν − 2ν

x2 − 2(ν + 1)Φ2

ν + 2νtΦν − 2ν2

Φ2ν + 2tΦν − 2ν

P5(x; t) = x5 + 2(ν + 2)tΦ2

ν + ν(2t2 − 1)Φν − 2ν2t

(ν + 1)Φ2ν + 2νtΦν − 2ν2

x3

+2[2t2 − (ν + 1)2

]Φ2ν − 4ν(ν + 3)tΦν + 4(ν + 2)ν2

(ν + 1)Φ2ν + 2νtΦν − 2ν2

x

where

Φν(t) = 2t−√

2D1−ν

(√2 t)

D−ν(√

2 t)

Theorem (PAC, Jordaan & Kelil [2015])Suppose that the monic polynomialsQn(x; t) are generated by the three-term

recurrence relation

xQn(x; t) = Qn+1(x; t) + 14[1− (−1)n]tQn−1(x; t),

with Q0(x; t) = 1 and Q1(x; t) = x and the monic polynomials Pn(x; t) arisefrom the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

)Then

Q2n(x; t) = (x2 − 12t)

n

Q2n+1(x; t) = x(x2 − 12t)

n

and in the limit as t→∞P2n(x; t)→ (x2 − 1

2t)n = Q2n(x; t)

P2n+1(x; t)→ x(x2 − 12t)

n = Q2n+1(x; t)

This is due to the fact that for the generalized Freud weight, as t→∞β2n(t)→ 0, β2n+1(t)→ 1

2t

Theorem (PAC, Jordaan & Kelil [2015])For the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

)the monic orthogonal polynomials Pn(x; t) satisfy the differential-differenceequation

dPndx

(x; t) = −Bn(x; t)Pn(x; t) + An(x; t)Pn−1(x; t)

where

An(x; t) = 4(x2 − 1

2t + βn + βn+1

)βn

Bn(x; t) = 4xβn +(2ν + 1)[1 + (−1)n+1]

2x

with βn(t) the recurrence coefficient in the three-term recurrence relation

Pn+1(x; t) = xPn(x; t)− βn(t)Pn−1(x; t)

Theorem (PAC, Jordaan & Kelil [2015])For the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

)the monic orthogonal polynomials Pn(x; t) satisfy the differential-differenceequation

dPndx

(x; t) = −Bn(x; t)Pn(x; t) + An(x; t)Pn−1(x; t)

where

An(x; t) = 4(x2 − 1

2t + βn + βn+1

)βn

Bn(x; t) = 4xβn +(2ν + 1)[1 + (−1)n+1]

2x

with βn(t) the recurrence coefficient in the three-term recurrence relation

Pn+1(x; t) = xPn(x; t)− βn(t)Pn−1(x; t)

For Hermite polynomials Hn(x) and Laguerre polynomials L(α)n (x):

d

dxHn(x) = 2nHn−1(x),

d

dxL(α)n (x) =

n

xL(α)n x)− n + α

xL

(α)n−1(x)

Theorem (PAC, Jordaan & Kelil [2015])For the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

)the monic orthogonal polynomials Pn(x; t) satisfy the differential equation

d2Pn

dx2 (x; t) + Rn(x; t)dPndx

(x; t) + Tn(x; t)Pn(x; t) = 0

where

Rn(x; t) = −4x3 + 2tx− 2ν + 1

x− 2x

x2 − 12t + βn + βn+1

Tn(x; t) = 4nx2 + 4βn + 16(βn + βn+1 − 12)(βn + βn−1 − 1

2)βn

− 8x2βn + (2ν + 1)[1 + (−1)n+1]

x2 − 12t + βn + βn+1

+ (2ν + 1)[1 + (−1)n+1]

(t− 1

2x2

)with βn(t) the recurrence coefficient in the three-term recurrence relation

Pn+1(x; t) = xPn(x; t)− βn(t)Pn−1(x; t)

Theorem (PAC, Jordaan & Kelil [2015])For the generalized Freud weight

ω(x; t) = |x|2ν+1 exp(−x4 + tx2

)the monic orthogonal polynomials Pn(x; t) satisfy the differential equation

d2Pn

dx2 (x; t) + Rn(x; t)dPndx

(x; t) + Tn(x; t)Pn(x; t) = 0

where

Rn(x; t) = −4x3 + 2tx− 2ν + 1

x− 2x

x2 − 12t + βn + βn+1

Tn(x; t) = 4nx2 + 4βn + 16(βn + βn+1 − 12)(βn + βn−1 − 1

2)βn

− 8x2βn + (2ν + 1)[1 + (−1)n+1]

x2 − 12t + βn + βn+1

+ (2ν + 1)[1 + (−1)n+1]

(t− 1

2x2

)with βn(t) the recurrence coefficient in the three-term recurrence relation

Pn+1(x; t) = xPn(x; t)− βn(t)Pn−1(x; t)

For Hermite and Laguerre polynomials:

d2Hn

dx2 − 2xdHn

dx+ 2nHn = 0,

d2L(α)n

dx2 +α + 1− n

x

dL(α)n

dx+n

xL(α)n = 0

Orthogonal polynomials on complex contours

ω(x; t) = exp(−1

3x3 + tx

), t > 0

• PAC, A Loureiro & W Van Assche, “Unique positive solution for thealternative discrete Painlevé I equation", arXiv:1508.04916 (2015).

Consider the semi-classical Airy weight

ω(x; t) = exp(−1

3x3 + tx

), t > 0

on the curve C from e2πi/3∞ to e−2πi/3∞. The moments are

µ0(t) =

∫C

exp(−1

3x3 + tx

)dx = Ai(t)

µk(t) =

∫Cxk exp

(−13x

3 + tx)

dx =dk

dtkAi(t) = Ai(k)(t)

where Ai(t) is the Airy function, the Hankel determinant is

∆n(t) =W(Ai(t),Ai′(t), . . . ,Ai(n−1)(t))

with ∆0(t) = 1, and the recursion coefficients are

αn(t) =d

dtln

∆n+1(t)

∆n(t)=

d

dtlnW(Ai(t),Ai′(t), . . . ,Ai(n)(t)

)W(Ai(t),Ai′(t), . . . ,Ai(n−1)(t)

)βn(t) =

d2

dt2ln ∆n(t) =

d2

dt2lnW(Ai(t),Ai′(t), . . . ,Ai(n−1)(t)

)with

α0(t) =d

dtln Ai(t) =

Ai′(t)Ai(t)

, β0(t) = 0

The recurrence coefficients αn(t) and βn(t) satisfy the discrete system

(αn + αn−1)βn − n = 0

α2n + βn + βn+1 − t = 0

(1)

and the differential system (Toda)dαndt

= βn+1 − βn, dβndt

= βn(αn − αn−1) (2)

Eliminating αn−1 and βn+1 between (1) and (2) yieldsdαndt

= −αn − 2βn + t,dβndt

= 2αnβn − n (3)

Letting xn = −βn and yn = −αn in (1) and (2) yieldsxn + xn+1 = y2

n − txn(yn + yn−1) = n

(4)

anddxndt

= xn(yn−1 − yn),dyndt

= xn+1 − xn (5)

Eliminating xn+1 and yn−1 between (4) and (5) yieldsdyndt

= y2n − 2xn − t, dxn

dt= −2xnyn + n (6)

Consider the systemdyndt

= y2n − 2xn − t

dxndt

= −2xnyn + n

• Eliminating xn yieldsd2yn

dt2= 2y3

n − 2tyn − 2n− 1

which is equivalent tod2q

dz2 = 2q3 + zq + n + 12

i.e. PII with A = n + 12.

• Eliminating yn yieldsd2xn

dt2=

1

2xn

(dxndt

)2

+ 4x2n + 2txn − n2

2xnwhich is equivalent to

d2v

dz2 =1

2v

(dv

dz

)2

− 2v2 − zv − n2

2v

an equation known as P34.

The Airy solutions of the equationsd2yn

dt2= 2y3

n − 2tyn − 2n− 1, y0(t) = −Ai′(t)Ai(t)

d2xn

dt2=

1

2xn

(dxndt

)2

+ 4x2n + 2txn − n2

2xn, x0(t) = 0

are

yn(t) =d

dtln

τn(t)

τn+1(t)=

d

dtlnW(Ai(t),Ai′(t), . . . ,Ai(n−1)(t)

)W(Ai(t),Ai′(t), . . . ,Ai(n)(t)

)xn(t) = − d2

dt2ln τn(t) = − d2

dt2lnW(Ai(t),Ai′(t), . . . ,Ai(n−1)(t)

)where

τn(t) =W(Ai(t),Ai′(t), . . . ,Ai(n−1)(t)), n ≥ 1

and τ0(t) = 1.

As t→∞yn(t) = t1/2 +

2n + 1

4 t− 12n2 + 12n + 5

32 t5/2+

(2n + 1)(16n2 + 16n + 15)

64 t4+O(t−11/2)

xn(t) =n

2 t1/2− n2

4 t2+

5n(4n2 + 1)

64 t7/2− n2(8n2 + 7)

16 t5+O(t−13/2)

xn + xn+1 = y2n − t (1a)

xn(yn + yn−1) = n (1b)Solving (1b) for xn and substituting in (1a) yields

n + 1

yn + yn+1+

n

yn + yn−1= y2

n − t (2)

which is known as alt-dPI (Fokas, Grammaticos & Ramani [1993]).Consider the Bäcklund transformations

yn+1 = −yn +2(n + 1)

y2n + y′n − t

(3a)

yn−1 = −yn +2n

y2n − y′n − t

(3b)

Eliminating y′n yields alt-dPI (2), whilst letting n→ n + 1 in (3b) and substi-tuting (3a) yields

d2yn

dt2= 2y3

n − 2tyn − 2n− 1 (4)

which is equivalent to PII [yn(t) = −21/3q(z), t = −2−1/3z, with α = n + 12].

Remark: The system (1) can also be written as

xn+1 = y2n − xn − t, yn+1 = −yn +

n + 1

y2n − xn − t

Theorem (PAC, Loureiro & Van Assche [2015])For positive values of t, there exists a unique solution of

xn + xn+1 = y2n − t

xn(yn + yn−1) = n

with x0(t) = 0 for which xn+1(t) > 0 and yn(t) > 0 for all n ≥ 0. This solutioncorresponds to the initial value

y0(t) = −Ai′(t)Ai(t)

.

Theorem (PAC, Loureiro & Van Assche [2015])For positive values of t, there exists a unique solution of

xn + xn+1 = y2n − t

xn(yn + yn−1) = n

with x0(t) = 0 for which xn+1(t) > 0 and yn(t) > 0 for all n ≥ 0. This solutioncorresponds to the initial value

y0(t) = −Ai′(t)Ai(t)

.

Theorem (PAC, Loureiro & Van Assche [2015])For positive values of t, there exists a unique solution of

n + 1

yn + yn+1+

n

yn + yn−1= y2

n − t

for which yn(t) ≥ 0 for all n ≥ 0. This solution corresponds to the initialvalues

y0(t) = −Ai′(t)Ai(t)

, y1(t) = −y0(t) +1

y20(t)− t

Lemma If 0 < t1 < t2 thenyn(t1) < yn(t2), xn(t1) > xn(t2)

i.e. yn(t) is monotonically increasing and xn(t) is monotonically decreasing.Lemma For fixed t with t > 0 then

√t < yn(t) < yn+1(t),

1

2√t>xn(t)

n>xn+1(t)

n + 1

yn(t), n = 1,5,10,15,20 1nxn(t), n = 1,5,10,15,20

Question: What happens if we don’t require that t > 0?

yn(t) = − d

dtln

τn(t)

τn+1(t), xn(t) = − d2

dt2ln τn(t), τn(t) =

[dj+k

dtj+kAi(t)

]n−1

j,k=0

yn(t), n = 1,2,3,4 1nxn(t), n = 1,2,3,4

Question: What happens if we have a linear combination of Ai(t) and Bi(t)?

y0(t; θ) = − d

dtlnϕ(t; θ), x1(t; θ) = − d2

dt2lnϕ(t; θ)

ϕ(t; θ) = cos(θ) Ai(t) + sin(θ) Bi(t)

y0(t; θ) x1(t; θ)

θ = 0, 11000

π, 1100π, 1

25π, 1

10π, 1

5π, 1

2π

y1(t; θ) y2(t; θ) y3(t; θ)

12x2(t; θ) 1

3x3(t; θ) 14x4(t; θ)

Conclusions• The coefficients in the three-term recurrence relations associated with

semi-classical generalizations of orthogonal polynomials can often be ex-pressed in terms of solutions of the Painlevé equations.• These recursion coefficients can be expressed as Hankel determinants

which arise in the solution of the Painlevé equations, in particular thePainlevé σ-equations, the second-order, second-degree equations associ-ated with the Hamiltonian representation of the Painlevé equations.• These Hankel determinants arise in the special cases of the Painlevé

equations when they have solutions in terms of the classical special func-tions, the “classical solutions” of the Painlevé equations.• The moments of the semi-classical weights provide the link between the

orthogonal polynomials and the associated Painlevé equation.• These ideas can be extended to orthogonal polynomials in other contexts:∗ discrete orthogonal polynomials (PAC [2013]); and∗ orthogonal polynomials on the unit circle (PAC & Smith [2015]).

• These results illustrate the increasing significance of the Painlevé equa-tions in the field of orthogonal polynomials and special functions.

References

• P A Clarkson, “Recurrence coefficients for discrete orthonormal polyno-mials and the Painlevé equations", J. Phys. A 46 (2013) 185205• P A Clarkson & K Jordaan, “The relationship between semi-classical

Laguerre polynomials and the fourth Painlevé equation", Constr. Approx.,39 (2014) 223–254• P A Clarkson, K Jordaan & A Kelil, “A Generalized Freud Weight",

preprint (2015).• P A Clarkson, A F Loureiro & W Van Assche, “Unique positive so-

lution for the alternative discrete Painlevé I equation", arXiv:1508.04916(2015).• J G Smith & P A Clarkson, “The fifth Painlevé equation and semi-

classical orthogonal polynomials", preprint (2015).• J G Smith & P A Clarkson, “The fifth Painlevé equation and orthogonal

polynomials associated with discontinuous weights", preprint (2015).

14th International Symposium on “OrthogonalPolynomials, Special Functions and Applications"

University of KentCanterbury, UK

3rd-7th July 2017

For further information see

http://www.kent.ac.uk/smsas/personal/opsfa/

Examples Associated with the Fifth Painlevé Equation

d2q

dz2 =

(1

2q+

1

q − 1

)(dq

dz

)2

− 1

z

dq

dz+

(q − 1)2

z2

(Aq +

B

q

)PV

+Cq

z+Dq(q + 1)

q − 1(z

d2σ

dz2

)2

=

[2

(dσ

dz

)2

− zdσ

dz+ σ

]2

− 4

4∏j=1

(dσ

dz+ κj

)SV

The Kummer functions M(a, b, z) and U(a, b, z) have the integral represen-tations

M(a, b, z) =Γ(b)

Γ(a)Γ(b− a)

∫ 1

0

ezs sa−1(1− s)b−a−1 ds

U(a, b, z) =1

Γ(a)

∫ ∞0

e−zs sa−1(1 + s)b−a−1 ds

• For the perturbed Jacobi weight (Basor, Chen & Ehrhardt [2010])ω(x; z) = xα−1(1 + x)β−1e−zx, x ∈ [0, 1], α > 0, β > 0

the moments are given by

µ0(z;α, β) =Γ(α)Γ(β)

Γ(α + β)e−zM(α, α + β, z), µk(z;α, β) = (−1)k

dk

dzkµ0(z;α, β)

• For the Pollaczek-Jacobi weight (Chen & Dai [2010])ω(x; z) = xα−1(1− x)β−1e−z/x, x ∈ [0, 1], α > 0, β > 0

the kth moment isµk(z;α, β) = Γ(β) e−zU(β, 1− α− k, z)

For both these weights Hn(z) = zd

dzln ∆n(z), with ∆n(z) = det

[µj+k(z)

]n−1

j,k=0,

satisfies an equation which is equivalent to a special case of SV, the PV σ-equation.

Theorem (Okamoto [1987]; Forrester & Witte [2002])Special function solutions of the PV σ-equation(

zd2σ

dz2

)2

−{

2

(dσ

dz

)2

− zdσ

dz+ σ

}2

+ 4

4∏j=1

(dσ

dz+ κj

)= 0 SV

are given by

σ(z) = zd

dzln τn(ϕα,β)− 1

4(3n + 2α− β − 1)z

− 58n

2 − 14(2α− 3β − 1)n− 1

8(2α− β − 1)2

for the parametersκ1 = 1

4(2α− β − n− 1), κ3 = 14(2α− β + 3n− 1)

κ2 = −14(2α + β + n− 3), κ4 = −1

4(2α− 3β + n + 1)

where τn(ϕα,β) is the determinant given by

τn(ϕα,β) = det

[(z

d

dz

)j+kϕα,β(z)

]n−1

j,k=0

withϕα,β(z) = C1M(α, β, z) + C2U(α, β, z)

C1 and C2 arbitrary constants, M(α, β, z) and U(α, β, z) Kummer functions.

Deformed Laguerre weightConsider orthogonal polynomials with the respect to the deformed La-

guerre weightω(x; z) = xν(x + z)λe−x, x ∈ R+, ν > 0, λ > 0

Define the Hankel determinant

∆n(z; ν, λ) = det[µj+k(z; ν, λ)

]n−1

j,k=0

where

µk(z; ν, λ) =

∫ ∞0

xν+k(x + z)λe−x dx

which can be evaluated in terms of the Kummer function U(a, b, z). Chen& McKay [2012] (also Basor, Chen & McKay [2013]) show that

Hn(z; ν, λ) = zd

dzln ∆n(z; ν, λ)

satisfies(z

d2Hn

dz2

)2

=

[(z + 2n + ν + λ)

dHn

dz−Hn + nλ

]2

− 4dHn

dz

(dHn

dz+ λ

)[z

dHn

dz−Hn + n(n + ν + λ)

]which is equivalent to a special case of SV, the PV σ-equation.

Remarks• For the deformed Laguerre weight

ω(x; z) = xν(x + z)λe−x, x ∈ R+, ν > 0, λ > 0

the kth moment is

µk(z; ν, λ) =

∫ ∞0

xν+k(x + z)λe−x dx

= zν+λ+k+1

∫ ∞0

sν+k(1 + s)λe−sz dx

= Γ(ν + k + 1)zν+λ+k+1U(ν + k + 1, ν + λ + k + 2, t)

with U(a, b, z) the Kummer function of the second kind.• In the special case of the deformed Laguerre weight when λ = m ∈ Z+

then

µk(z; ν,m) =

∫ ∞0

xν+k(x + z)me−x dx

= Γ(ν + k + 1)zν+m+k+1U(ν + k + 1, ν + m + k + 2, t)

= Γ(ν + k + 1) (−1)mm!L(−ν−m−k−1)m (z)

with L(α)n (z) the Laguerre polynomial, sincezα+mU(α, α + m + 1, z) = (−1)mm!L(−α−m)

m (z), m ∈ Z+

Discontinuous Laguerre weight(PAC & Smith)

Consider the discontinuous Laguerre weight

ω(x; z) = {1− ξH(x− z)} |x− z|λ xν exp(−x), ν, λ > 0, x, z ∈ R+

with H(x) the Heaviside step function.Since∫ z

0

xν(z − x)λe−x dx = B(λ + 1, ν + 1) zν+λ+1 e−zM(λ + 1, ν + λ + 2, z)∫ ∞z

xν(x− z)λe−x dx = Γ(λ + 1) zν+λ+1 e−z U(λ + 1, ν + λ + 2, z)

withB(a, b) = Γ(a)Γ(b)/Γ(a+b) the Beta function, andM(a, b, z) and U(a, b, z)the Kummer functions, then

µ0(z; ν, λ) =

∫ ∞0

[1− ξH(x− z)]xν |x− z|λ e−x dx

=

∫ z

0

xν(z − x)λ e−x dx + (1− ξ)

∫ ∞z

xν(x− z)λ e−x dx

= zν+λ+1 e−z {B(λ + 1, ν + 1)M(λ + 1, ν + λ + 2, z)

+ (1− ξ)Γ(λ + 1)U(λ + 1, ν + λ + 2, z)}

Define the Hankel determinant

∆n(z; ν, λ) = det[µj+k(z; ν, λ)

]n−1

j,k=0

thenHn(z; ν, λ) = z

d

dzln ∆n(z; ν, λ)

satisfies

z2

(d2Hn

dz2

)2

=

[(z + 2n + ν + λ)

dHn

dz−Hn + (2n + 2ν + λ)n

]2

− 4

(dHn

dz+ n

)(dHn

dz+ n + ν

)[z

dHn

dz−Hn + (n + ν + λ)n

]which is equivalent to a special case of SV, the PV σ-equation. Specifically,letting

Hn(z; ν, λ) = σ − 14(2n + ν − λ)z + 1

2n2 + 1

2n(ν + λ) + 18(ν − λ)2

yields (z

d2σ

dz2

)2

−{

2

(dσ

dz

)2

− zdσ

dz+ σ

}2

+ 4

4∏j=1

(dσ

dz+ κj

)= 0 SV

withκ1 = 1

2n + 34ν + 1

4λ, κ3 = −12n− 1

4ν − 34λ

κ2 = 12n− 1

4ν + 14λ κ4 = −1

2n− 14ν + 1

4λ