A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Pacharoni, I.; Román, Pablo

doi:10.1007/s00365-007-0673-1

Cited by 42 publications

(21 citation statements)

References 13 publications

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“…This is very similar to the analysis in section 3.1 of [11], where the final result is equations (3)- (5), expressing these polynomials in terms of the matrix valued hypergeometric function 2 F 1 of Tirao; see [35]. As mentioned in the last section, such an expression in terms of 2 F 1 is obtained in [28] for a family of polynomials that are related to {Q n (x)} n≥0 in the form …”

Section: The New Equivalent Classical Pair Let Us Consider the Follosupporting

confidence: 63%

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Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Grünbaum¹,

Iglesia²

2008

SIAM J. Matrix Anal. & Appl.

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Abstract. We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (SU(N + 1), U(N )); see [I. Pacharoni and J. A. Tirao, Constr. Approx., 25 (2007), pp. 177-192]. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which we can explicitly compute its "nstep transition probability matrix" and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.

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Section: The New Equivalent Classical Pair Let Us Consider the Follosupporting

confidence: 63%

“…In [28] one finds an explicit expression for a family of eigenfunctions of D in terms of the matrix valued hypergeometric function 2 F 1 introduced in [35].…”

Section: A Family Of Examples Arising From the Complex Projective Spacementioning

confidence: 99%

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Grünbaum¹,

Iglesia²

2008

SIAM J. Matrix Anal. & Appl.

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“…If we replace α by m ∈ Z ≥0 and β by n − 1 ∈ Z, this differential equation arises from the first few steps in the explicit determination of all matrix valued spherical functions associated to the n-dimensional projective space P n (C) = SU(n+1)/U(n). Furthermore, the solutions of (3.3) can be used to construct an explicit expression of a family of matrix valued orthogonal polynomials with respect to a certain weight matrix W (u) (see [1,2]). …”

Section: An Example From Representation Theorymentioning

confidence: 99%

The Generalized Matrix Valued Hypergeometric Equation

Román

Simondi

2010

Int. J. Math.

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The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

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“…As it happens for the more studied subject of orthogonal matrix polynomials which are eigenfunctions of second-order differential operators (see [4][5][6][7][8][9][10][11][12][13][14] and the references therein), the examples mentioned above have allowed to discover many new and certainly interesting phenomena which are absent in the well-known scalar theory of classical discrete orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 96%

The convex cone of weight matrices associated to a second-order matrix difference operator

Durán

Ríos

2014

Integral Transforms and Special Functions

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We associate to a given finite order difference operator D with matrix coefficients the convex cone ϒ(D) formed by all weight matrices W with respect to which the operator D is symmetric. In the scalar case, the convex cone of positive measures associated to a second-order difference operator always reduces to the empty set except for those operators associated to the classical discrete families of Charlier, Meixner, Krawtchouk or Hahn, in which case the convex cone is the half line defined by the classical discrete measure itself. In the matrix case the situation is rather different. We develop two methods to study these convex cones and, using them, we construct some illustrative examples of second-order difference operators whose convex cones are, at least, two dimensional.

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A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Cited by 42 publications

References 13 publications

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

The Generalized Matrix Valued Hypergeometric Equation

The convex cone of weight matrices associated to a second-order matrix difference operator

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