Matrix valued orthogonal polynomials of the Jacobi type

Grünbaurn, F.A.; Pacharoni, I.; Tirao, Juan

doi:10.1016/s0019-3577(03)90051-6

Cited by 72 publications

(104 citation statements)

References 19 publications

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“…There has been quite a bit of work in terms of producing explicit examples of these polynomials. For a representative sample see [3,6,7,11,15,16,17,33,34]. Most of this work deals with looking for examples that would enjoy the appropriate bispectral property.…”

Section: Some Closing Challengesmentioning

confidence: 99%

The Rahman Polynomials Are Bispectral

Grünbaum¹

2007

SIGMA

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Abstract. In a very recent paper, M. Rahman introduced a remarkable family of polynomials in two variables as the eigenfunctions of the transition matrix for a nontrivial Markov chain due to M. Hoare and M. Rahman. I indicate here that these polynomials are bispectral. This should be just one of the many remarkable properties enjoyed by these polynomials. For several challenges, including finding a general proof of some of the facts displayed here the reader should look at the last section of this paper. Dedicated to the memory of Vadim Kuznetsov I first met Vadim at a meeting in Esterel, Canada back in 1994. I only saw him a few more times after that, in Amsterdam, in Leeds, in Louvain-la-Neuve, and the last time in Leganes. Although we never wrote any papers together we shared a number of common interests. I always found Vadim extremely helpful and willing to share his ideas and his vast knowledge. He was also willing to put up with people whose work was not in the mainstream and he usually would have some useful remark or a suggestion to make. I like to think that he would have enjoyed seeing this paper which builds on some remarkable work of another common friend I met for the first time at the same Esterel meeting. PreliminariesStarting about thirty years ago, M. Hoare and M. Rahman, published a number of papers studying a class of statistical models that were nicely associated with "classical" orthogonal polynomials in one variable. The reader can consult [19,20,21,4].In a very recent paper [22], which I have seen before publication thanks to the kindness of M. Rahman, they take a huge step and consider the case of several variables. I propose to call these statistical models with the name Hoare-Rahman, and to call the multivariable polynomials that arise with the name Rahman polynomials. I am thankful to Prof. Rahman for several inspiring conversations on the contents of [22], in different coffee houses and restaurants in Fremont, California on the last days of 2006.The model of interest here captures very well the idea of a game of chance where the player takes a risk that can never improve his/her winnings but is then given a second opportunity to improve his/her lot. I am convinced that models of this kind should be of great interest in several areas of applied mathematics not only in physics (where they were conceived) but also in biology, population dynamics, evolutionary models, etc. On top of their potential for applications the polynomials discovered by M. Rahman are a piece of beauty. I hope that this

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Section: Some Closing Challengesmentioning

confidence: 99%

The Rahman Polynomials Are Bispectral

Grünbaum¹

2007

SIGMA

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“…In [2], the study of the matrix valued orthogonal polynomials which are eigenfunctions of certain second order differential operators was started. The first explicit examples of such polynomials are given in [7], [8] and [3].…”

Section: Introductionmentioning

confidence: 99%

“…In [8] (see also [3]) it is proved that the condition of symmetry for D is equivalent to the following three differential equations…”

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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“…The following proposition establishes the connection between the matrix L defined in (20) and the matrix R defined in (9).…”

Section: Notementioning

confidence: 95%

Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory

Miranian¹

2005

J. Phys. A: Math. Gen.

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In the work presented below the classical subject of orthogonal polynomials on the real line is discussed in the matrix setting. An analogue of the determinant definition of orthogonal polynomials is presented; the classical properties such as the recurrence relation, the kernel polynomials, and the Christoffel-Darboux formula are discussed. A τ -function for the system of matrix-valued orthogonal polynomials on the real line is presented. Some properties of the τ -functions are investigated.

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Matrix valued orthogonal polynomials of the Jacobi type

Cited by 72 publications

References 19 publications

The Rahman Polynomials Are Bispectral

The Rahman Polynomials Are Bispectral

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory

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