Su2,diag explicit form of the matrix valued qdifference operators of askeywilson type to which the matrix valued orthogonal polynomials are eigenfunctions. Author links open overlay panel wolter groenevelt a mourad e. A hypergeometric function transform and matrixvalued. Matrix valued orthogonal polynomials of the jacobi type. Pdf matrix valued orthogonal polynomials arising from. As far as we know, the matrixvalued orthogonal polynomials that we obtain have not been considered before. We consider a matrixvalued spectral decomposition of a family of blocktridiagonal matrices arising as the transition matrices of socalled quasibirthanddeath processes. The two periodic aztec diamond and matrix valued orthogonal. Matrixvalued orthogonal polynomials related to the quantum. Some very important results of the theory of scalar valued orthogonal polynomials, like favards theorem and markovs theorem have been extended to the matrix valued case, see 10, 11, 12, 21, 16, and many more still. Tirao department of mathematics, university of california, berkeley ca 94705, usa email. In preparing this paper, we have used computer algebra in order to verify the state. One way to understand them is to consider the discretization of the inner product of l2a,b.
However, in its sequel 9 the authors devoted a entire chapter to the construction of monogenic homogeneous polynomials that are orthogonal with respect to the inner product 1. Matrix case d can be noncommutative, have more than one generator and have relations among the generators. Matrixvalued orthogonal polynomials related to the. Matrix valued orthogonal polynomials satisfying di. Matrixvalued orthogonal polynomials on the real line berkeley math. Again, no qmatrixvalued orthogonal polynomials have yet emerged from this approach. A matrix recurrence for systems of cli ord algebravalued. Matrixvalued generalizations of the jacobi polynomials and of multipleorthogonal polynomials.
The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials p n n, with respect to a weight w, that are eigenfunctions of a second order di. The analytic theory of matrix orthogonal polynomials. Matrix valued orthogonal polynomials arising from group. Dedicated to tom koornwinder on the occasion of his 60th birthday. Left and right orthogonal matrix polynomials on the real line are closely related since p,x r p,xl t, which follows from p, ql r hence for orthogonal matrix polynomials on the real line it is sufficient to treat only the left orthogonal matrix polynomials. Computation of matrixvalued formally orthogonal polynomials. Matrixvalued gegenbauertype polynomials springerlink. In this article we utilize the moments of the orthogonality measure to represent matrix valued orthogonal polynomials on the unit circle as. At the same time the matrix approach sheds new light on results about systems of clifford algebravalued orthogonal polynomials obtained by gurlebeck, bock. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block jacobi. In this article we utilize the moments of the orthogonality measure to represent matrix valued orthogonal polynomials on the unit circle as certain schur complements. Matrix valued orthogonal polynomials have been introduced and studied by krein in connection with spectral analysis and moment problems, see 8, 33, 34.
Hermann schulzbaldes developed a sturm oscillation theory for matrix orthogonal polynomials on the real line in 10. Jul 17, 2019 hermitetype matrix valued weights are studied in detail. Matrix valued orthogonal polynomials have applications in the study of higherorder recurrences and their spectral analysis see, e. Matrix valued laguerre polynomials are introduced via a matrix weight function involving several degrees of freedom using the matrix nature. The analytic theory of matrix orthogonal polynomials david damanik, alexander pushnitski, and barry simon january 30, 2008 abstract we survey the analytic theory of matrix orthogonal polynomials. Spectral theory for matrix orthogonal polynomials on the unit.
In most of these papers, the matrix valued orthogonal polynomials are eigenfunctions of a second order matrix valued differential operator, so that these polynomials are usually considered as. Matrix valued orthogonal polynomials on the real line in this chapter the classical notion of scalar valued orthogonal polynomials on the real line is extended to matrix valued ones. At the same time the matrix approach sheds new light on results about systems of cli ord algebra valued orthogonal polynomials obtained by gurleb eck, bock, l avi cka, delanghe et al. In fact a larger class of random walks, known under the name of. Matrix valued orthogonal polynomials arising from group representation theory and a family of quasibirthanddeath processes. With the highly developed riemannhilbert toolkit at hand, we may thus hope to compute the asymptotic behavior of the polynomials, and more importantly the correlation kernel. Orthogonal matrixvalued polynomials and applications seminar on operator theory at the school of mathematical sciences, tel aviv university. However, except for special cases such as square orthogonal matrix polynomials on the line 25, the theory of the associated matrixvalued polynomials is a lot less developed than in the case of scalar moments. The matrix valued hypergeometric functions have connections with the theory of matrix valued spherical functions and with the theory of matrix valued orthogonal polynomials 2, 3, 4.
Matrix valued orthogonal polynomials satisfying differential. Given a nontrivial matrixvalued measure on the unit circle d, we can define. Matrix valued orthogonal polynomials of the jacobi type dedicuted to tom koornwinder on the occasion of his 60fh birthday by f. In 17, it had been shown that matrix valued orthogonal polynomials on the real line satisfy a noncommutative version of the toda equations, and in 6 one of the authors provided an analogue result for matrix valued orthogonal polynomials. The monic matrix valued orthogonal polynomials pn are expressed in terms of tiraos matrix valued hypergeometric function using the matrix valued differential operators of first and second order. Pdf a hypergeometric function transform and matrix. Jun 21, 2016 for these matrix valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation.
In 8, this technique has been used to deduce a matrix version of the discrete painlev e i equation, while. Tirao an invitation to matrix valued spherical functions. A matrix recurrence for systems of clifford algebravalued. Among the generalizations of the theory of orthogonal polynomials is the theory of matrix valued orthogonal polynomials in one variable. Spectral theory for matrix orthogonal polynomials on the. Presented below is an explicit matrix expression for the scalarmatrix valued orthogonal polynomials on the unit circle in terms of the moments of the measure. Matrixvalued orthogonal polynomials arose in the work of krein 26, 27 and have been studied from an analytic point of view by duran and others, see 610, 14, 16, 17 and references given there. Pdf a hypergeometric function transform and matrixvalued. Orthogonal matrix valued polynomials and applications seminar on operator theory at the school of mathematical sciences, tel aviv university.
In this paper, we discuss a new way to obtain matrixvalued orthogonal polynomials with an explicit threeterm recurrence relation as well as explicit orthogonality relations. Given a family of matrix valued orthogonal polynomials that are common eigenfunctions of one di. Matrix valued orthogonal polynomials, jacobi polynomials. Matrix valued orthogonal polynomials and quantum groups. This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal cliffordalgebra valued polynomials of arbitrary degree. Finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrixvalued functions.
Bochners problem as mentioned before, in 1929 bochner characterized all families of scalar orthogonal polynomials satisfying second order di. In the study of matrix valued orthogonal polynomials the general theory deals with obtaining appropriate analogues of classical results known for scalar valued orthogonal polynomials, and many. Matrix valued orthogonal polynomials on the unit circle. Ladder relations for a class of matrix valued orthogonal. Sep 25, 2014 this paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal cliffordalgebra valued polynomials of arbitrary degree. In this case we need polynomials with values in the set of square matrices of size two, whereas up to now we were dealing with scalar valued polynomials. In the study of matrixvalued orthogonal polynomials the general theory deals with obtaining appropriate analogues of classical results known for scalarvalued orthogonal polynomials, and many. Matrix valued orthogonal polynomials for gelfand pairs of. Matrix valued orthogonal polynomials arising from the. Matrixvalued orthogonal polynomials related to su2. Matrix valued orthogonal polynomials arising from the complex. In addition, we apply matrixvalued orthogonal polynomial techniques to the study of singraphs and higherdimensional birthanddeath processes, for which the relevant polynomials are multivariate. Orthogonal matrixvalued polynomials and applications. Generalization of the notion of orthogonality for jacobi polynomials in the case of general weights.
Under suitable conditions on the parameters the matrix weight function satisfies matrix pearson equations, which allow to introduce shift operators for these polynomials. An introduction to orthogonal polynomials marek rychlik. In 6,19 the subject of matrix valued orthogonal polynomials on the unit circle was approached from the point of view of minimization problem. Spectral decomposition and matrixvalued orthogonal. Spectral decomposition and matrixvalued orthogonal polynomials. The point of view is to construct the matrixvalued orthogonal polynomials using matrixvalued spherical functions, and next using this group theoretic interpretation to obtain properties of the matrixvalued orthogonal polynomials. Aug 29, 20 using jacobi polynomials, the operator t can also be realized as a fivediagonal operator, leading to orthogonality relations for 2. Section 6 concerns the matrixvalued version of the kernel polynomials and the christoffeldarboux formula. The concept of formally orthogonal polynomials can be extended to the case of arbitrary, in general rectangular, matrix valued moments. We extend to matrixvalued stochastic processes, some wellknown relations between realvalued diffusions and classical orthogonal polynomials, along with some recent results about levy processes and martingale polynomials. Again, no q matrix valued orthogonal polynomials have yet emerged from this approach.
The algebra of di erential operators that is obtained in this way is denoted by d. Matrix valued orthogonal polynomials math berkeley. These matrixvalued polynomials can be considered as matrixvalued generalizations of wilson polynomials. Ladder relations for a class of matrix valued orthogonal polynomials. For these matrixvalued orthogonal polynomials, a number of properties are derived using this quantum group interpretation. Pdf matrix valued orthogonal polynomials arising from group. He used the results of symplectic geometry developed by bott in 7, which connects the spectra of a matrix to the intersection of lagrangian planes. In most of these papers, the matrixvalued orthogonal polynomials are eigenfunctions of a second order matrixvalued differential operator, so that these polynomials are usually considered as. However, except for special cases such as square orthogonal matrix polynomials on the line 25, the theory of the associated matrix valued polynomials is a lot less developed than in the case of scalar moments. Orthogonal polynomials 75 where the yij are analytic functions on c \ r, and solve for such matrices the following matrixvalued riemannhilbert problem. Twoperiodic aztec diamond and matrix valued orthogonal polynomials arno kuijlaars ku leuven joint work with maurice duits kth stockholm preprint is on arxiv on monday. Volume 14, issues 34, 15 december 2003, pages 353366. Orthogonal matrix polynomials and applications core. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra.
The matrixvalued spherical functions give rise to matrixvalued orthogonal polynomials, which are matrixvalued analogues of a subfamily of askeywilson polynomials. The family of matrix valued orthogonal polynomials p n x. The space cx endv is a bimodule over the matrix algebra endv. Matrixvalued orthogonal polynomials date back to the 1950s in the work of m. Matrix valued orthogonal polynomials related to compact. Xiii brunelbielefeld workshop on random matrix theory zif, bielefeld, germany, 16 december 2017. The key fact is that these matrix valued orthogonal polynomials can be characterized in terms of the solution of a 2p 2pmatrix valued riemannhilbert problem. Matrix valued polynomials are elements in cx endv where v is a nite dimensional complex vector space. Luisa miranian matrix valued orthogonal polynomials. The concept of formally orthogonal polynomials can be extended to the case of arbitrary, in general rectangular, matrixvalued moments.
Matrixvalued orthogonal polynomials have been introduced and studied by krein in connection with spectral analysis and moment problems, see 8, 33, 34. This representation is a generalization of the karlinmcgregor representation for the nstep transition probabilities of a birthanddeath process via a system of orthogonal polynomials. Matrixvalued orthogonal polynomials on the real line. Matrix valued orthogonal polynomials of the jacobi type dedicated to tom koornwinder on the occasion of his 60th birthday author links open overlay panel f. Twoperiodic aztec diamond and matrix valued orthogonal. Matrixvalued orthogonal polynomials have applications in the study of higherorder recurrences and their spectral analysis see, e. A matrix recurrence for systems of cli ord algebravalued orthogonal polynomials i. Using recent results by casper and yakimov, we investigate mvops. Orthogonal polynomials in statistics the polynomials commonly used as orthogonal contrasts for quantitative factors are discrtete analogues of legendre polynomials. The monic matrixvalued orthogonal polynomials pn are expressed in terms of tiraos matrixvalued hypergeometric function using the matrixvalued differential operators of first and second order. Matrix valued orthogonal polynomials related to compact gel. Matrix valued orthogonal polynomials of jacobi type. In this paper, we discuss a new way to obtain matrix valued orthogonal polynomials with an explicit threeterm recurrence relation as well as explicit orthogonality relations.
892 1103 249 7 457 819 408 865 1187 71 237 1324 1074 518 336 1263 1488 969 486 975 1123 847 1413 606 1324 168 50 1324 791 1391 1345 734 978 1300 940 847 836 1250 1176 360 60 209 412 72 1 325 1065 357