On certain properties of a perturbed Freud-type weight Abey Kelil - - PowerPoint PPT Presentation

On certain properties of a perturbed Freud-type weight

Abey Kelil University of Pretoria AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications Douala, Cameroon October 09, 2018

1

Plan of the talk

1 Introduction 2 Background 3 A class of perturbed Freud type weight

A ‘Generalized Freud’ Weight

4 Conclusions & Future perspectives

2

Introduction

Orthogonal polynomials on the real line Let P = span{xk : k ∈ N0} be the linear space of polynomials with real coefficients and consider the inner product ·, ·µ : P × P → R: f , gµ = b

a

f (x)g(x) dµ(x), f , g ∈ P, supp(µ) = [a, b] ⊆ R. Let {Pn(x)}∞

n=0 be a monic orthogonal polynomial sequence with respect to

this inner product: Pm, Pnµ = b

a

Pn(x)Pm(x) dµ(x) = ζnδmn, ζn > 0. If µ is absolutely continuous; i.e., dµ(x) = w(x) dx, then µk = b

a

xkw(x) dx < ∞, k = 0, 1, 2, . . . , are said to be moments for a positive weight function w(x).

3

Introduction

Theorem (Three-term recurrence) Let {Pn(x)}∞

n=0 be a sequence of monic orthogonal polynomials on [a, b] relative

to an inner product ·, ·w, P0(x) = 1, P1(x) = x − xP0(x), P0(x)w P0(x), P0(x)w . Then {Pn(x)}∞

n=0 satisfies the recursive scheme

xPn(x) = Pn+1(x) + αnPn(x) + βn Pn−1(x), (1) where αn = xPn, Pn Pn2 = 1 ζn

• R

x P2

n(x)w(x) dx; βn =

1 ζn−1

• R

x Pn(x) Pn−1w(x) dx > 0. Given the positive measure µ (the weight w), what are the recurrence coefficients? For classical orthogonal polynomials (Jacobi, Hermite, Laguerre), their recurrence coefficients are explicit (cf. Chihara, Szeg´

• , Rainville).

4

Background

Semi-classical orthogonal polynomials Classical orthogonal polynomials are characterized by their weight function, which satisfy Pearson’s equation [σ(x)w(x)]

′

= τ(x)w(x), (2) where σ, τ ∈ P with deg(σ) ≤ 2 and deg(τ) = 1 and boundary conditions (σw) (a) = 0 = (σw) (b), whereas semi-classical orthogonal polynomials have (2) with deg(σ) > 2 or deg(τ) = 1 (Hendriksen and van Rossum, 1977).

Weight function w(x) Parameters σ(x) τ(x) Semi-classical Laguerre xλexp(−x2 + tx) λ > −1 x 1 + λ + tx − 2x2 Freud exp(− 1

4x4 − tx2)

x, t ∈ R 1 −2tx − x3 Generalized Freud |x|2λ+1exp(−x4 + tx2) λ > 0, x, t ∈ R x 2λ + 2 − 2tx2 − x4

These polynomials satisfy a structural relation (Maroni, 1985)

σ(x)P

′

n+1(x) = n+r

• j=n−s

An,jPj(x),

• r = deg(σ),

s = max{deg(σ) − 2, deg(τ) − 1}

.

In this case, the recurrence coefficients are usually not explicit and they obey non-linear recurrence relations.

5

Background

Certain semi-classical weights obey non-linear recurrence Equations The weight w(x) = exp(−x4) on R (cf. Nevai, 1983): Since αn = 0 (symmetry) in the ttrr (1), the coefficient βn obeys 4βn (βn−1 + βn + βn+1) = n; β0 = 0, β1 = ∞

−∞ x2 exp(−x4) dx

∞

−∞ exp(−x4) dx

= Γ( 3

4)

Γ( 1

4).

The semiclassical Laguerre wν(x) = xν exp(−x2 + tx), ν > −1, x ∈ R+ : (W. Van Assche, L. Boelen (2011) and ( P. Clarkson, K. H. Jordaan (2014)) (2αn − t) (2αn−1 − t) = (2βn − n) (2βn − n − ν) βn , 2βn + 2βn+1 − αn(2αn − t) = 2n + 1 + ν.    The Freud weight w(x) = exp

• −x4 + tx2

, x ∈ R (Freud (1976)): n βn = −2t + 4 [βn+1 + βn + βn−1] , β0 = 0, (3) (3) is known as Shohat-Freud’s (‘String’ or Discrete Painlev´ e) equation. Asymptotic behavior: βn = n

12

1/2 1 +

1 24n2 + O(n−4)

• ( Nevai, 1984).

6

Background

The link to Painlev´ e equations

Some history: The first non-linear recurrence equation - Shohat (1930’s) and Laguerre, Freud (late 70’s) and very recently recognized as discrete Painlev´ e equations by Fokas, Its, and Kitaev. Work by Magnus (relation between discrete and continuous Painlev´ e equations), Witte, Clarkson, Van Assche, Nijhoff, Spicer, Chen and Ismail extended theory with some more examples. The Painlev´ e equations are a chapter in ‘DLMF’.

d2q dt2 = 6q2 + z, z ∈ C (PI); PII(α) : d2q dz2 = 2q3 + zq + α, PIII(α, β, γ, δ) : d2q dz2 = 1 q

dq

dz

2

− 1 z dq dz + 1 z (αq2 + β) + γq3 + δ q , PIV(α, β) : d2q dz2 = 1 2q

dq

dz

2

+ 3 2 q3 + 4zq2 + 2(z2 − α)q + β q , (4)

where α, β, γ, δ are constants and z ∈ C. Some discrete Painlev´ e equations: (d-PI) xn+1 + xn + xn−1 = zn + γ(−1)n xn + σ (d-PII) xn+1 + xn−1 = xnzn + γ 1 − x 2

n

(d-PIV) (xn+1 + xn) (xn + xn−1) =

• x 2

n − κ2

x 2

n − µ2

(xn + zn)2 − γ2

7

Background

Certain semi-classical weights giving rise to (discrete) Painlev´ e equations

w(x) = |x|̺ exp(−x 4), ̺ > −1 on R is related to dPI (Magnus, 1986). w(x; t) = exp(− 1

4x 4 + tx 2), t ∈ R on R is related to dPI (Magnus, 1995).

In the continuous sense, w(x) = x λ exp(−x 2 + tx) on R+ is related to PIV (Galina et. al, Clarkson et.al). w(x, t) = (1 − x)α(1 + x)β exp(−tx), α, β > −1, x ∈ [−1, 1], t ∈ R related to PV, (Basor, Chen and Ehrhardt (2009)). w(x, t) = x α exp(−x − t/x), α > −1, x ∈ R+, related to PIII, Chen & Its (2010). Some observations Solutions of Painlev´ e are sometimes not directly the recurrence coefficients, but functions of these, with extra terms and/or changes of variable. For e.g. (after Clarkson & Jordaan), if w(x; t) = x λ exp −x 2 + tx , λ > −1 then the function qn(z) = 2αn(t) + t, with z = 1

2t satisfies PIV in z &

(A, B) = (2n + λ + 1, −2λ2). The solutions obtained from deforming OPs are typically not ‘generic’, but with very specific values of the parameters. Special function solutions of PIV are expressed in terms of parabolic cylinder functions.

8

Background

Ladder relations help to obtain nonlinear equations

Assume the weight w vanishes at the endpoints of [a, b] ⊆ R. The ladder operators for the polynomials Pn(z) (cf. Chen & Ismail, Van Assche et.al) are given by

d

dz + Bn(z)

Pn(z) = βnAn(z)Pn−1(z)

d

dz − Bn(z) − ν′(z)

Pn−1(z) = −An−1(z)Pn(z)

• (5)

where ν(x) = − log w(x), since w(x) > 0, x ∈ [a, b] ⊆ R and the coefficients in (5) are given by An(z) = 1 hn

∞

−∞

P2

n(y)

• ν′(z) − ν′(y)

z − y

• w(y) dy,

Bn(z) = 1 hn−1

∞

−∞

Pn(y)Pn−1(y)

• ν′(z) − ν′(y)

z − y

• w(y) dy,

Note: We can calculate An(z) and Bn(z) without explicitly knowing the polynomials

• ther than the weight function.

9

Background

Compatibility conditions (cf. Chen & Ismail, Magnus, Van Assche et.al), For the ladder operators, the associated compatibility conditions are Lemma The functions An(z) and Bn(z) satisfy Bn+1(z) + Bn(z) = (z − αn) An(z) − ν′(z) 1 + (z − αn) (Bn+1(z) − Bn(z)) = βn+1An+1(z) − βnAn−1(z), valid for all z ∈ C ∪ {∞}. The functions An(z), Bn(z) and

n−1

• k=0

Ak(z) satisfy the identity B2

n(z) + ν′(z)Bn(z) + n−1

• k=0

Ak(z) = βnAn(z)An−1(z).

10

A class of perturbed Freud type weight

Extract from Digital Library of Mathematical Functions

§18.32 Orthogonal polynomials 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, non-negative, and continuously differentiable. Of special interest are the cases Q(x) = x2m, m = 1, 2, . . . . No explicit expressions for the corresponding OP’s are available. However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky [2001]. For a uniform asymptotic expansion in terms of Airy functions for the OP’s in the case x4 see Bo and Wong [1999]. Our interest: Can we obtain concise formulations for the recurrence coefficients as well as the polynomials that are orthogonal with respect to the generalized Freud weight? What more non-linear recurrence relations can one obtain?

11

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

A ‘Generalized Freud’ Weight

Monic orthogonal polynomials with respect to the generalized Freud inner product p, q =

• R

p(x)q(x)|x|2λ+1 exp(−x 4 + tx 2) dx, x ∈ R, (t ∈ R), λ > 0, (7) satisfy the three-term recurrence relation xSn(x; t) = Sn+1(x; t) + βn(t; λ)Sn−1(x; t), n = 1, 2, . . . , (8) where initial conditions S−1 ≡ 0, S0 ≡ 1 and βn(t; λ) > 0. The coefficients βn(t; λ) satisfy a Toda-type equation as in below. Theorem (Aptekarev et. al) Let µ be a symmetric positive measure on R for which all the moments exist and let µt be the measure for which dµt(x) = exp(tx 2) dµ(x), where t ∈ R is such that all the moments of µt exist. Then the recurrence coefficients of the orthogonal polynomials for µt satisfy the differential-difference equation d dt βn = βn [βn+1 − βn−1] , n = 1, 2, . . . . (9)

12

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Moments for the generalized Freud weight Lemma (The first moment) The first moment, µ0(t; λ), for generalized Freud weight is given by µ0(t; λ) = Γ(λ + 1) 2(λ+1)/2 exp 1

8t2

D−λ−1

• − 1

2

√ 2 t

• ,

where Dν(ξ) = exp(− 1

4ξ2)

Γ(−ν) ∞ s−ν−1 exp

• − 1

2s2 − ξs

• ds,

Re(ν) < 0, is a parabolic cylinder (Hermite-Weber) function. This result follows from the definition of the first moment µ0(t; λ) = ∞

−∞

|x|2λ+1 exp

• −x4 + tx2

dx and the integral representation of the parabolic cylinder function.

13

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Higher-order moments

The even moments are µ2n(t; λ) =

∞

−∞

x 2n |x|2λ+1 exp −x 4 + tx 2 dx ≡ µ0(t; λ + n) = dn dtn

∞

−∞

|x|2λ+1 exp −x 4 + tx 2 dx

• = dn

dtn µ0(t; λ), n = 1, 2, . . . , whilst the odd ones are µ2n+1(t; λ) =

∞

−∞

x 2n+1 |x|2λ+1 exp −x 4 + tx 2

• Odd

dx = 0, n = 1, 2, . . . , since the integrand is odd. When λ = n ∈ Z+, D−n−1

• −1

2 √ 2t

• = 1

2 √ 2π dn dtn 1 + erf(1 2t)

• exp(1

8t2)

• ,

with erfc(z) is the error function.

14

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Recurrence Coefficients (Concise form)

Theorem (Concise formulation of βn(t; λ) in terms of Tau function) The recurrence coefficients {βn(t; λ)}∞

n=1 in the three-term recurrence relation

xSn(x; t) = Sn+1(x; t) + βn(t; λ)Sn−1(x; t), n ∈ N, are explicitly given by β2n(t; λ) = d dt ln τn(t; λ + 1) τn(t; λ) ; β2n+1(t; λ) = d dt ln τn+1(t; λ) τn(t; λ + 1) , (10) where τn(t; λ) is the Wronskian given by τn(t; λ) = W

• φλ, d φλ

dt , . . . , dn−1φλ dtn−1

• = det
• dj+k

dtj+k µ0(t; λ)

n−1

j,k=0

, with τ0(t; λ) = 1 and φλ(t) = µ0(t; λ) =

Γ(λ+1) 2(λ+1)/2 exp 1 8 t2

D−λ−1

• − 1

2

√ 2 t . Remark: The function Hn(t; λ) := d

dt ln τn(t; λ) satisfies the 2nd order, 2nd degree equation

4

• d2Hn

dt2

2

−

• t d Hn

dt − Hn

2

+ 4 d Hn dt

• 2 d Hn

dt − n

• 2 d Hn

dt − n − λ

• = 0.

which is equivalent to SIV, the PIV σ-equation and so the coefficients take the form β2n(t; λ) = Hn(t; λ + 1) − Hn(t; λ); β2n+1(t; λ) = Hn+1(t; λ) − Hn(t; λ + 1).

15

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Sample recurrence coefficients in terms of Φλ The first few recurrence coefficients βn(t; λ) are given by β1(t; λ) = Φλ, β2(t; λ) = −2Φ2

λ − tΦλ − λ − 1

2Φλ , β3(t; λ) = − Φλ 2Φ2

λ − tΦλ − λ − 1 − λ + 1

2Φλ , where Φλ(t) = d dt ln

• D−λ−1
• − 1

2

√ 2 t

• exp

1

8t2

= 1

2t + 1 2

√ 2 D−λ

• − 1

2

√ 2 t

• D−λ−1
• − 1

2

√ 2 t .

16

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

The first few polynomials By using the recurrence relation xSn(x; t) = Sn+1(x; t) + βn(t; λ)Sn−1(x; t), with S0(x; t) = 1, S0(x; t) = 0, β0(t; λ) = 0, then the first few polynomials are given by S1(x; t) = x, S2(x; t) = x2 − Φλ, S3(x; t) = x3 − tΦλ + λ + 1 2Φλ x, S4(x; t) = x4 − 2tΦ2

λ − (t2 + 2)Φλ − (λ + 1)t

2(2Φ2

λ − tΦλ − λ − 1)

x2 − 2(λ + 2)Φ2

λ − (λ + 1)tΦλ − (λ + 1)2

2(2Φ2

λ − tΦλ − λ − 1)

.

17

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Linking βn(t; λ) to the general dPI Theorem (Link of the coefficient βn(t; λ) to dPI) The recurrence coefficients associated with the generalized Freud weight wλ(x; t) satisfy the nonlinear difference equation; i.e., the general discrete dPI: βn+1 + βn + βn−1 = 1

2t + 2n + (2λ + 1)[1 − (−1)n]

8βn , where β0 = 0 and β1 is given by β1(t; λ) = µ2(t; λ) µ0(t; λ) = ∞

−∞ x2|x|2λ+1 exp

• −x4 + tx2

dx ∞

−∞ |x|2λ+1 exp (−x4 + tx2) dx

= 1

2t + 1 2

√ 2 D−λ

• − 1

2

√ 2 t

• D−λ−1
• − 1

2

√ 2 t .

18

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Sketch of the proof: Recall that ζn = β1 · · · βn. Consider the integral Jn :=

• R

[Pn−1(y)Pn(y)]

′wλ(y)dy.

By using orthogonality, we obtain Jn = nζn−1. Integrating by parts yields

Jn = nζn−1 =

• R
• 4y 3 − 2ty − 2λ + 1

y

• Pn(y)Pn−1(y)wλ(y)dy

=

• R

[4y 2 − 2t] [Pn+1(y) + βnPn−1(y)] Pn−1(y)wλ(y)dy −

• R

2λ + 1 y Pn(y)Pn−1(y)wλ(y)dy = −2tβnζn−1 + 4ζn+1 + 4βn

• R

(Pn(y) + βn−1Pn−1(y))2wλ(y)dy − γΩnζn−1 = −2tβnζn−1 + 4ζn+1 + 4βn[ζn + β2

n−1ζn−2] − γΩnζn−1,

γ = 2λ + 1.

This yields dPI.

19

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Linking βn(t; λ) to continuous PIV

Theorem The coefficient βn(t; λ) in equation (8) satisfy the nonlinear differential equation d2βn dt2 = 1 2βn

d βn

dt

2

+ 3

2 β3 n − tβ2 n + ( 1 8 t2 − 1 2 An)βn +

Bn 16βn , (11) which is equivalent to PIV, where the parameters An and Bn are given by

A2n

B2n

• =

−2λ − n − 1

−2n2

• ;

A2n+1

B2n+1

• =
• λ − n

−2(λ + n + 1)2

• .

Remark: Equation (11) ≡ PIV through βn(t; λ) = 1

2 ν(z), with z = − 1 2 t. Hence

β2n(t; λ) = 1

2 ν

z; −2λ − n − 1, −2n2 , β2n+1(t; λ) = 1

2 ν

z; λ − n, −2(λ + n + 1)2 , (12) with z = − 1

2 t, where ν(z; A, B) satisfies PIV.

Sketch of the proof: Consider the dPI with βn = xn; i.e., n + (2λ + 1)Ωn = 4xn

• xn+1 + xn + xn−1 − 1

2 t

• ,

(13) Combine (13) with the Toda equation (9); that is, x

′

n = xn(xn+1 − xn−1),

and finally eliminate xn+1 and xn−1 to obtain PIV as in (11).

20

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

Ladder relations for the case generalized Freud Theorem The monic orthogonal polynomials Sn(x; t) with respect to the generalized Freud weight on R satisfy the differential-difference recurrence relation S′

n(x; t) = βnAn(x; t)Sn−1(x; t) − Bn(x; t)Sn(x; t)

(14) where An(x; t) = 4x2 + 4(βn + βn+1) − 2t, Bn(x; t) = 4xβn + 2λ + 1 x

• Ωn,

where the expression Ωn is given by Ωn = 1 − (−1)n 2 . Equation (14) is also known as Structural relation or Lowering operators.

21

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

(More) non-linear Difference equations

Theorem (Non-linear equations) For the weight in (7), the coefficients βn in (8) satisfy the following system of equations: βn+1

• 4βn+2 + 4βn+1 − 2t
• − βn
• 4βn+1 + 4βn − 2t
• = (2λ + 1) [Ωn+1 − Ωn] + 1,

n−1

• k=0

(βk + βk+1) = 16β2

n

• (βn−1 + βn + βn+1 − t) + [(8βn+1 − 4t) (2βn−1 − t)]

+ 2t [n + (2λ + 1)Ωn + n]

• ,

Theorem (Differential-recurrence) For the generalized Freud weight, the recurrence coefficients βn = βn(t) satisfy d βn dt = βn (βn+1 − βn−1) , d2βn dt2 = βn

• βn+1 (βn+1 + βn+2 − 2βn−1 − βn) + βn−1 (βn−2 + βn−1 − βn)
• .

22

A class of perturbed Freud type weight A ‘Generalized Freud’ Weight

The differential equation

Theorem For the generalized Freud weight, the monic orthogonal polynomials Sn(x; t) satisfy the differential equation x d2Sn dx 2 (x; t) + Rn(x; t)dSn dx (x; t) + Tn(x; t)Sn(x; t) = 0, where the coefficients are given by Rn(x; t) = x

• −4x 3 + 2tx + 2λ + 1

x − 2x x 2 − 1

2t + βn + βn+1

• ,

Tn(x; t) = x 4nx 2 + 4βn + 16βn(βn + βn+1 − t

2)(βn + βn−1 − t 2)

− 8βnx 2 + (2λ + 1)[1 − (−1)n] x 2 − 1

2t + βn + βn+1

+ 4(2λ + 1)(−1)nβn +(2λ + 1)[1 − (−1)n]

• t −

1 2x 2

• .

Note that one can find a similar result for Freud-like weights (cf. A. Arceo, E.J. Huertas & F. Marcell´ an (2016)).

23

Conclusions & Future perspectives

Conclusions The recurrence coefficients associated with the generalized Freud weight can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the PIV equation. The moments of the generalized Freud weight provide the link between the

• rthogonal polynomials and the associated Painlev´

e equation. A concise formulation of the generalized Freud polynomials has also been obtained. Part of the results of this work illustrate the increasing significance of the Painlev´ e equations in the field of orthogonal polynomials and special functions. Future perspectives Investigation of a class of polynomials orthogonal with respect to a more general Shohat-Freud type weight function. Certain properties for polynomials orthogonal with respect to a perturbed Airy-type and Hexic-Freud weights are under investigation: (NB: The weight exp −x 3 + tx (’deformed Airy type’) as a modification of exp(−x 3) with an exponential factor exp(tx) is considered by Van Assche, Filipuk and Zhang (2015)).

24

Conclusions & Future perspectives

Thank you!

25