The matrix-valued hypergeometric equation

Tirao, Juan

doi:10.1073/pnas.1337650100

Cited by 82 publications

(90 citation statements)

References 9 publications

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“…, 0. 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: 64%

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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: 64%

“…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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“…The matrix valued hypergeometric function was studied in [15]. Let V be a d-dimensional complex vector space, and let A, B and C ∈ End(V ) ; z F 0 is a solution of the hypergeometric equation (1) such that F (0) = F 0 .…”

Section: Introductionmentioning

confidence: 99%

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Pacharoni

Román

2007

Constr Approx

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Abstract. The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {Pn}n, with respect to a weight W , that are eigenfunctions of a second order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.

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“…In this construction, one obtains a family of matrix-valued functions Ψ n , together with a matrix-valued differential operator Ω, for which the functions Ψ n are eigenfunctions. The first of these functions, Ψ 0 , turns out to be invertible, and the sequence P n = Ψ n Ψ −1 0 is a sequence of matrix-valued orthogonal polynomials with respect to an appropriate weight function which are eigenfunctions of a matrix-valued hypergeometric operator as in [39].…”

Section: Introductionmentioning

confidence: 99%

Some bivariate stochastic models arising from group representation theory

Iglesia

Román

2018

Stochastic Processes and their Applications

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Abstract. The aim of this paper is to study some continuous-time bivariate Markov processes arising from group representation theory. The first component (level) can be either discrete (quasi-birth-and-death processes) or continuous (switching diffusion processes), while the second component (phase) will always be discrete and finite. The infinitesimal operators of these processes will be now matrix-valued (either a block tridiagonal matrix or a matrix-valued secondorder differential operator). The matrix-valued spherical functions associated to the compact symmetric pair (SU(2) × SU(2), diag SU(2)) will be eigenfunctions of these infinitesimal operators, so we can perform spectral analysis and study directly some probabilistic aspects of these processes. Among the models we study there will be rational extensions of the one-server queue and Wright-Fisher models involving only mutation effects.

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The matrix-valued hypergeometric equation

Cited by 82 publications

References 9 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

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Some bivariate stochastic models arising from group representation theory

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