New examples of matrix orthogonal polynomials satisfying second order differential equations. Structural formulas Joint work with A.J. Dur´ an University of Doc-course,University of Seville, spring 2010
Introduction Given a self-adjoint positive definite matrix valued weight function W ( t ) (of dimension N × N ) 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 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. Considering possible applications of MOP it is natural to concentrate on those cases where some extra property holds.
The Matrix Bochner’s Theorem In the nineties, A. Dur´ an takes a step in this direction raising 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
The matrix Bochner Problem 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 , Orthogonal Matrix Polynomials satisfying differential equations Int. Math Res. Not. 2004.
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
MOP related to second order differential op. Consider the recurrence relation for the sequence MOP ( P n ( t )) n ≥ 0 , orthonormal with respect to a weight matrix W 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. The examples of MOP satisfying second order differential equations existing up to now in the literature satisfy that A n B n lim ϕ ( n ) = aI, lim ϕ ( n ) = bI, n →∞ n →∞ for a convenient continuous function ϕ ( t ) . The example given here satisfies the property that the previous limits give no longer a scalar matrix
The method to find MOP satisfying 2 nd order differential eq.s (Dur´ n-Gr¨ unbaum), 2004 Simmetry Eqs : differential equations for the weight function W and the coefficients of ℓ 2 ,R = D 2 F 2 ( t ) + D 1 F 1 ( t ) + D 0 F 0 ( t ) . Symmetry Equations WF ∗ F 2 W = 2 2( F 2 W ) ′ F 1 W + WF ∗ = 1 ′′ − ( F 1 W ) ′ ( WF ∗ ( F 2 W ) = 0 − F 0 W ) We put W ( t ) = T ( t ) T ∗ ( t )
The talk continues according to the following suggestions: To carefully present the N × N example and the differential equation. Then to show the 2 × 2 case. Mention the method follow to obtain the generalization (the Theorem 2.3 of the paper [A. Dur´ an, Constructive Approx, 2009]) To present the Rodrigues formula, mentioning how it was obtained, that is, the differential equation for R n which appeared in [A. Dur´ an, Int. Math. Research Notices, 2010] Finally the recurrence relation for the simplest normalization and the recurrence for the orthonormal polynomials, remarking the facts that the limits of the recurrence coefficients are not scalar.
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