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
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. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P ( t ) and Q ( t ) by the numerical matrix � P ( t ) W ( t ) Q ∗ ( t ) dt, � P, Q � = � P, Q � W = R where Q ∗ ( t ) denotes the conjugate transpose of Q ( t ) . There exists a sequence ( P n ) n of matrix polynomials, orthonormal with respect to W and with P n of degree n . The sequence ( P n ) n is unique up to a product with a unitary matrix.
Introduction A new family Rodrigues formula Recurrence relations Matrix orthogonal polynomials Let W be an N × N positive definite matrix of measures. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P ( t ) and Q ( t ) by the numerical matrix � P ( t ) W ( t ) Q ∗ ( t ) dt, � P, Q � = � P, Q � W = R where Q ∗ ( t ) denotes the conjugate transpose of Q ( t ) . There exists a sequence ( P n ) n of matrix polynomials, orthonormal with respect to W and with P n of degree n . The sequence ( P n ) n is unique up to a product with a unitary matrix.
Introduction A new family Rodrigues formula Recurrence relations Matrix orthogonal polynomials Let W be an N × N positive definite matrix of measures. Consider the skew symmetric bilinear form defined for any pair of matrix valued functions P ( t ) and Q ( t ) by the numerical matrix � P ( t ) W ( t ) Q ∗ ( t ) dt, � P, Q � = � P, Q � W = R where Q ∗ ( t ) denotes the conjugate transpose of Q ( t ) . There exists a sequence ( P n ) n of matrix polynomials, orthonormal with respect to W and with P n of degree n . The sequence ( P n ) n is unique up to a product with a unitary matrix.
Introduction A new family Rodrigues formula Recurrence relations Matrix orthogonal polynomials Property Any sequence of orthonormal matrix valued polynomials ( P n ) n satisfies a three term recurrence relation A ∗ n P n − 1 ( t ) + B n P n ( t ) + A n +1 P n +1 ( t ) = tP n ( t ) , where P − 1 is the zero matrix and P 0 is non singular. A n are nonsingular matrices and B n hermitian.
Introduction A new family Rodrigues formula Recurrence relations The matrix Bochner’s problem In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations . Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying ′′ ′ P n ℓ 2 ,R = P n F 2 ( t ) + P n F 1 ( t ) + P n F 0 ( t ) = Λ n P n ( t ) , n ≥ 0 Right hand side differential operator ℓ 2 ,R = D 2 F 2 ( t ) + D 1 F 1 ( t ) + D 0 F 0 ( t ) . P n eigenfunctions, Λ n eigenvalues: P n ℓ 2 ,R = Λ n P n
Introduction A new family Rodrigues formula Recurrence relations The matrix Bochner’s problem In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations . Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying ′′ ′ P n ℓ 2 ,R = P n F 2 ( t ) + P n F 1 ( t ) + P n F 0 ( t ) = Λ n P n ( t ) , n ≥ 0 Right hand side differential operator ℓ 2 ,R = D 2 F 2 ( t ) + D 1 F 1 ( t ) + D 0 F 0 ( t ) . P n eigenfunctions, Λ n eigenvalues: P n ℓ 2 ,R = Λ n P n
Introduction A new family Rodrigues formula Recurrence relations The matrix Bochner’s problem In the nineties, A. Dur´ an formulated the problem of characterizing MOP which satisfy second order differential equations . Dur´ an, Rocky Mountain J. Math (1997) Characterize all families of MOP satisfying ′′ ′ P n ℓ 2 ,R = P n F 2 ( t ) + P n F 1 ( t ) + P n F 0 ( t ) = Λ n P n ( t ) , n ≥ 0 Right hand side differential operator ℓ 2 ,R = D 2 F 2 ( t ) + D 1 F 1 ( t ) + D 0 F 0 ( t ) . P n eigenfunctions, Λ n eigenvalues: P n ℓ 2 ,R = Λ n P n
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 there exists a non singular matrix T independent of t such that W ( t ) = TD ( t ) T ∗ with D ( t ) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2 nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.
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 there exists a non singular matrix T independent of t such that W ( t ) = TD ( t ) T ∗ with D ( t ) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2 nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.
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 there exists a non singular matrix T independent of t such that W ( t ) = TD ( t ) T ∗ with D ( t ) a diagonal matrix of weights The first examples of MOP, which does not reduce to scalar, satisfying 2 nd order differential equations in the framework of the general theory of orthogonal polynomials appeared in Dur´ an-Gr¨ unbaum , Matrix Orthogonal Polynomials satisfying differential equations Int. Math Res. Not. 2004.
Introduction A new family Rodrigues formula Recurrence relations Dr. Jekyll Dur´ an’s course 0 0 · · · 0 a 1 0 0 a 2 · · · 0 A = a 1 , · · · a N ∈ C \ { 0 } . . . . . . . . . . . . . . . . . . . 0 0 0 · · · a N 0 0 0 0 0 2( N − 1) 0 2( N − 2) W ( t ) = e − t 2 e At e A ∗ t F 0 = A 2 − , ... 0 0 The weight matrix has a symmetric second order differential operator of the form � d � d � 2 � + (2 A − 2 It ) + F 0 dt dt
Introduction A new family Rodrigues formula Recurrence relations Dr. Jekyll Dur´ an’s course 0 0 · · · 0 a 1 0 0 a 2 · · · 0 A = a 1 , · · · a N ∈ C \ { 0 } . . . . . . . . . . . . . . . . . . . 0 0 0 · · · a N 0 0 0 0 0 2( N − 1) 0 2( N − 2) W ( t ) = e − t 2 e At e A ∗ t F 0 = A 2 − , ... 0 0 The weight matrix has a symmetric second order differential operator of the form � d � d � 2 � + (2 A − 2 It ) + F 0 dt dt
Introduction A new family Rodrigues formula Recurrence relations The goal We aim to: Present a new example of MOP , of dimension N × N , satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as Rodrigues formula Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations The goal We aim to: Present a new example of MOP , of dimension N × N , satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as Rodrigues formula Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations The goal We aim to: Present a new example of MOP , of dimension N × N , satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as Rodrigues formula Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations The goal We aim to: Present a new example of MOP , of dimension N × N , satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as Rodrigues formula Recurrence relations
Introduction A new family Rodrigues formula Recurrence relations The goal We aim to: Present a new example of MOP , of dimension N × N , satisfying second order differential equations with different interesting properties. To present some structural formulas for the case N = 2 such as Rodrigues formula Recurrence relations
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