Introduction A new family Rodrigues formula Recurrence relations New examples of matrix orthogonal polynomials satisfying second order differential equations. Structural formulas J. Borrego Morell (*) , joint work with A. Dur´ an (**) and M. Castro (**) (*) Carlos III University of Madrid (**) University of Seville IMUS Doc-course, University of Seville, March-May 2010
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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
Introduction A new family Rodrigues formula Recurrence relations
Outline
Introduction
A new family
Rodrigues formula
Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations
Outline
Introduction
A new family
Rodrigues formula
Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations
Outline
Introduction
A new family
Rodrigues formula
Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations
Outline
Introduction
A new family
Rodrigues formula
Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations
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.
Introduction A new family Rodrigues formula Recurrence relations
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.
Introduction A new family Rodrigues formula Recurrence relations
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.
Introduction A new family Rodrigues formula Recurrence relations
Matrix orthogonal polynomials
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.
Introduction A new family Rodrigues formula Recurrence relations
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
Introduction A new family Rodrigues formula Recurrence relations
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
Introduction A new family Rodrigues formula Recurrence relations
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
Introduction A new family Rodrigues formula Recurrence relations
The matrix Bochner’s problem
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
Introduction A new family Rodrigues formula Recurrence relations
The matrix Bochner’s problem
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
Introduction A new family Rodrigues formula Recurrence relations
The matrix Bochner’s problem
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