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Introduction A new family Rodrigues formula Recurrence relations

New examples of matrix orthogonal polynomialssatisfying second order differential equations.

Structural formulas

J. Borrego Morell(∗) , joint work with A. Duran(∗∗) and M.Castro(∗∗)

(∗) Carlos III University of Madrid(∗∗) University of Seville

IMUS Doc-course, University of Seville, March-May 2010

Outline

Introduction

A new family

Rodrigues formula

Recurrence relations

Outline

Introduction

A new family

Rodrigues formula

Outline

Introduction

A new family

Rodrigues formula

Outline

Introduction

A new family

Rodrigues formula

Matrix orthogonal polynomials

Let W be an N ×N positive definite matrix of measures. Considerthe skew symmetric bilinear form defined for any pair of matrixvalued functions P (t) and Q(t) by the numerical matrix

〈P,Q〉 = 〈P,Q〉W =∫

RP (t)W (t)Q∗(t)dt,

where Q∗(t) denotes the conjugate transpose of Q(t).

There exists a sequence (Pn)n of matrix polynomials, orthonormalwith respect to W and with Pn of degree n.

The sequence (Pn)n is unique up to a product with a unitarymatrix.

〈P,Q〉 = 〈P,Q〉W =∫

Property

Any sequence of orthonormal matrix valued polynomials (Pn)nsatisfies a three term recurrence relation

A∗nPn−1(t) +BnPn(t) +An+1Pn+1(t) = tPn(t),

where P−1 is the zero matrix and P0 is non singular.An are nonsingular matrices and Bn hermitian.

The matrix Bochner’s problem

In the nineties, A. Duran formulated the problem of characterizingMOP which satisfy second order differential equations.

Duran, Rocky Mountain J. Math (1997)Characterize all families of MOP satisfying

Pn`2,R = P′′nF2(t) + P

′nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

Right hand side differential operator

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

Pn eigenfunctions, Λn eigenvalues:

Pn`2,R = ΛnPn

Pn`2,R = P′′nF2(t) + P

′nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

Pn`2,R = ΛnPn

Pn`2,R = P′′nF2(t) + P

′nF1(t) + PnF0(t) = ΛnPn(t), n ≥ 0

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

Pn`2,R = ΛnPn

We will say that a weight W does not reduce to scalar if thereexists a non singular matrix T independent of t such that

W (t) = TD(t)T ∗

with D(t) a diagonal matrix of weightsThe first examples of MOP, which does not reduce to scalar,satisfying 2nd order differential equations in the framework of thegeneral theory of orthogonal polynomials appeared in

Duran-Grunbaum , Matrix Orthogonal Polynomials satisfyingdifferential equations Int. Math Res. Not. 2004.

W (t) = TD(t)T ∗

Dr. Jekyll Duran’s course

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

F0 = A2−

2(N − 1) 0

2(N − 2). . .

, W (t) = e−t2eAteA

The weight matrix has a symmetric second order differentialoperator of the form(

)(2A− 2It) + F0

Dr. Jekyll Duran’s course

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

F0 = A2−

2(N − 1) 0

2(N − 2). . .

, W (t) = e−t2eAteA

The weight matrix has a symmetric second order differentialoperator of the form(

)(2A− 2It) + F0

The goal

We aim to:

Present a new example of MOP, of dimension N ×N ,satisfying second order differential equations with differentinteresting properties.

To present some structural formulas for the case N = 2 suchas

Rodrigues formulaRecurrence relations

The goal

We aim to:

The goal

We aim to:

The goal

We aim to:

The goal

We aim to:

The method to find MOP satisfying 2nd order differentialeq.s

(Duran-Grunbaum), 2004 Simmetry Eqs: differentialequations for the weight function W and the coefficients of

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

Symmetry Equations

F2W = WF ∗2

2(F2W )′ = F1W +WF ∗1

(F2W )′′ − (F1W )′ = (WF ∗0 − F0W )

The method to find MOP satisfying 2nd order differentialeq.s

(Duran-Grunbaum), 2004 Simmetry Eqs: differentialequations for the weight function W and the coefficients of

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

Symmetry Equations

F2W = WF ∗2

2(F2W )′ = F1W +WF ∗1

(F2W )′′ − (F1W )′ = (WF ∗0 − F0W )

MOP related to second order differential op.

Consider the recurrence relation for the sequence MOP (Pn(t))n≥0,orthonormal with respect to a weight matrix W

where P−1 is the zero matrix and P0 is non singular.

The examples of MOP satisfying second order differential equationsexisting up to now in the literature satisfy that

limn→∞

Anϕ(n)

= aI, limn→∞

Bnϕ(n)

for a convenient continuous function ϕ(t).

The example given here satisfies the property that the previouslimits give no longer a scalar matrix

limn→∞

Anϕ(n)

= aI, limn→∞

Bnϕ(n)

limn→∞

Anϕ(n)

= aI, limn→∞

Bnϕ(n)

limn→∞

Anϕ(n)

= aI, limn→∞

Bnϕ(n)

General case

W (t) = TT ∗, T (t) = eAteDt2

where D is a diagonal matrix independent of t

− (v+1)2 0

− (N−1)(v+1)2(v+N−1)

− (N−1)(v+1)2(jv+N−1)

0 −12

where v ∈ (−1,∞) \ {0}

General case

where D is a diagonal matrix independent of t

− (v+1)2 0

− (N−1)(v+1)2(v+N−1)

− (N−1)(v+1)2(jv+N−1)

0 −12

where v ∈ (−1,∞) \ {0}

General case

2 ]−1∑j=0

(−1)j(

14(N − 1)

)j (2j + 1)j−1

)jA2j+1

A is the N ×N nilpotent matrix

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

General case

2 ]−1∑j=0

(−1)j(

14(N − 1)

)j (2j + 1)j−1

)jA2j+1

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

General case

2 ]−1∑j=0

(−1)j(

14(N − 1)

)j (2j + 1)j−1

)jA2j+1

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

General case

2 ]−1∑j=0

(−1)j(

14(N − 1)

)j (2j + 1)j−1

)jA2j+1

0 a1 0 · · · 00 0 a2 · · · 0. . . . . . . . . . . . . . . . . . .0 0 0 · · · aN0 0 0 0 0

a1, · · · aN ∈ C \ {0}

General case

`2,R = D2F2(t) +D1F1(t) +D0F0(t).

F2(t) = Ψ2 +[N

2 ]−1∑j=0

(−1)jv(

)j 2j + 1N − 1

(2j + 1)j−1

j!A2j+1t

1 0v+N−1N−1

. . .jv+N−1N−1

0 v + 1

General case

Using the Theorem 2.3 A.Duran, Constructive Approx, 2009 weobtain the expression for

F1(t) = −2(k + 1)tI +[N

2 ]−1∑j=0

)jΓ2j+1A

[N2 ]−1∑j=0

)jA2jvt

General case

jv+N−1N−1 0

. . .kv+N−1N−1

. . .(N−2)v+N−1

0. . .

General case

jv+N−1N−1 0

. . .kv+N−1N−1

. . .(N−2)v+N−1

0. . .

General case

For the matrix F0(t) we have

F0(t) = (v+1)Ψ0+[N

2 ]−1∑j=1

(−1)j+1 jj−1

(N − 1)j2j−1j!

)jΓ2jA

where Ψ0 is the diagonal matrix

2j. . .

0 2(N − 1)

Particular cases

For N = 2 we have

), D =

(−v+1

2 00 −1

(|a1|2e−t

2t2 + e−(v+1)t2 a1e

−t2t

a1e−t2t e−t

Particular cases

The second order differential equation that satisfy the MOP is

P′′n (t)

(1 a1vt0 v + 1

′n(t)

(−2(v + 1)t 2a1(v + 1)

0 −2(v + 1)t

Pn(t)(

0 00 2(v + 1)

)=(−2(v + 1)N 0

0 −2(v + 1)(N − 1)

)Pn(t)

Rodrigues formula

Property (A. Duran, Int. Math. Research Notices, 2009)

Fix n and consider the second order differential equation

(XF ∗2 )′′ − (X[F ∗1 + n(F ∗2 )

′])′+X

[F ∗0 + n(F ∗1 )

′+(n

)(F ∗2 )

′′]

Write Rn = X for a solution of this equation. Then the function

Pn(t) = R(n)n (t)W−1(t) satisfies

P′′n (t)F2(t) + P

′n(t)F1(t) + Pn(t)F0(t) = ΛnPn(t)

Rodrigues formula

Consider

0 a0 0

), D =

(− b

2 00 −1

(|a|2e−t2t2 + e−bt

2ae−t

ae−t2t e−t

Rn(t) = e−t2

(b−ne(1−b)t

2 (n+ 2t2) at

a(2t+ et2√π(1 + Erf(

√bt)− Erf(t)) 2

where Erf(t) =2√π

0e−x

Pn(t) = R(n)n (t)W−1(t), ||Pn||2 = 2n−1√πn!

(αn+1

0 4αn

αn = 2 + a2bn−12n, b = v + 1

Rodrigues formula

Consider

0 a0 0

), D =

(− b

2 00 −1

(|a|2e−t2t2 + e−bt

2ae−t

ae−t2t e−t

Rn(t) = e−t2

(b−ne(1−b)t

2 (n+ 2t2) at

√bt)− Erf(t)) 2

0e−x

Pn(t) = R(n)n (t)W−1(t), ||Pn||2 = 2n−1√πn!

(αn+1

0 4αn

αn = 2 + a2bn−12n, b = v + 1

Rodrigues formula

Consider

0 a0 0

), D =

(− b

2 00 −1

(|a|2e−t2t2 + e−bt

2ae−t

ae−t2t e−t

Rn(t) = e−t2

(b−ne(1−b)t

2 (n+ 2t2) at

√bt)− Erf(t)) 2

0e−x

Pn(t) = R(n)n (t)W−1(t), ||Pn||2 = 2n−1√πn!

(αn+1

0 4αn

αn = 2 + a2bn−12n, b = v + 1

The sequence of polynomials (Pn)n≥0 orthogonal with respect toW generated by the above Rodrigues formula satisfy the threeterm recurrence relation

tPn(t) = An+1Pn+1(t) +BnPn(t) + CnPn−1(t), n ≥ 1,

An = −12

(1 00 αn−1

Bn = a(−n+ (n+ 1)b)

2bn−12

αn+10

Cn = −n( αn+1

), αn = 2 + a2bn−

with the initial conditions P−1 = 0, P0 =(

1 00 2

The sequence of polynomials (Pn)n≥0 orthogonal with respect toW generated by the above Rodrigues formula satisfy the threeterm recurrence relation

tPn(t) = An+1Pn+1(t) +BnPn(t) + CnPn−1(t), n ≥ 1,

An = −12

(1 00 αn−1

Bn = a(−n+ (n+ 1)b)

2bn−12

αn+10

Cn = −n( αn+1

), αn = 2 + a2bn−

with the initial conditions P−1 = 0, P0 =(

1 00 2

tPn(t) = An+1Pn+1(t) + BnPn(t) + A∗nPn−1(t), , n ≥ 1,

Pn(t) orthonormal w.r.t. W (t)

An =√n

√αn+1√

2b√αn

0√αn−1√2√αn

ab2n−3

4 (b+ (1− b)n)√αnαn+1

(0 11 0

with the initial conditions P−1 = 0, P0 = (π)−14

( √2b

tPn(t) = An+1Pn+1(t) + BnPn(t) + A∗nPn−1(t), , n ≥ 1,

Pn(t) orthonormal w.r.t. W (t)

An =√n

√αn+1√

2b√αn

0√αn−1√2√αn

ab2n−3

4 (b+ (1− b)n)√αnαn+1

(0 11 0

with the initial conditions P−1 = 0, P0 = (π)−14

( √2b

Observe that

limn→∞

An√n

(1√2

00 1√

)if b > 1(

1√2b

00 1√

)for 0 < b < 1

That is, we have obtained an example where the limits ofAnϕ(n)

for certain convenient continuous function ϕ(x), give a diagonalmatrix and not a scalar matrix of the form cI, c ∈ R.However lim

n→∞Bn = 0.

Observe that

limn→∞

An√n

(1√2

00 1√

)if b > 1(

1√2b

00 1√

)for 0 < b < 1

n→∞Bn = 0.

Observe that

limn→∞

An√n

(1√2

00 1√

)if b > 1(

1√2b

00 1√

)for 0 < b < 1

n→∞Bn = 0.

new examples of matrix orthogonal polynomials satisfying ... · imus doc-course, university of...

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