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Semi-classical orthogonal polynomialsand the Painlevé equations
Peter A ClarksonSchool of Mathematics, Statistics and Actuarial Science
University of Kent, Canterbury, CT2 7NF, [email protected]
Analytic, Algebraic and Geometric Aspects of Differential EquationsBedlewo, Poland
14 September 2015
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Outline1. Introduction
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Outline1. Introduction2. Properties of the second Painlevé equation
d2q
dz2 = 2q3 + zq + α
•Hamiltonian structure• Bäcklund transformations and associated difference equations• Airy solutions
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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
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Outline1. Introduction2. Properties of the second Painlevé equation3. Properties of the fourth Painlevé equation4. Orthogonal polynomials
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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
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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∞.
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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
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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.
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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.
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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
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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
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• 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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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)
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Properties of the Second Painlevé Equation
d2q
dz2 = 2q3 + zq + α PII
• Hamiltonian structure• Bäcklund transformations and
associated difference equations• Airy solutions
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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.
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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]).
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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.
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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.
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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
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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
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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
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Airy Solutions of P34 pn(z; θ) =d2
dz2ln τn(z; θ)
Plots of p(z; 0)/n for n = 2,4,6,8
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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
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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
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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
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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
∞.
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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.
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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
)
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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)
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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.
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Orthogonal Polynomials
• Some History
• Monic orthogonal polynomials
• Semi-classical orthogonal polynomials
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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].
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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.
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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).
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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
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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.
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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
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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)
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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
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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
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• 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)
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• 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
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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
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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)
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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
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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)
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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).
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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, . . .
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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.
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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)
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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)
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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.
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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].
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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).
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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)
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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
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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)
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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)
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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)
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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
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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).
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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
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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)
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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.
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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)
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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
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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)
.
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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
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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
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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
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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
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y1(t; θ) y2(t; θ) y3(t; θ)
12x2(t; θ) 1
3x3(t; θ) 14x4(t; θ)
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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.
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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).
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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/
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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
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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.
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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.
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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.
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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+
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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)}
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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
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