The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: Five instructive examples

Castro, Mirta M.; Grünbaum, F. Alberto

doi:10.1155/imrn/2006/47602

Cited by 41 publications

(56 citation statements)

References 34 publications

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“…These two operators do not commute with each other. (c) As a final example we mention that in [CG2] one sees that for the Hermite-type polynomials introduced in [DG1], and further studied in [DG2], the space of second-order differential operators having them as common eigenfunctions has dimension four. This means that besides the equation given explicitly in [DG1], see (8.2), the monic orthogonal polynomials satisfy, for example, the equation…”

Section: Closing Remarksmentioning

confidence: 98%

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Durán¹,

Grünbaum²

2005

Journal of Approximation Theory

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Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) ). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues' formulas of the type ( n W ) (n) W −1 , where is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues' formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues' formula and show that scalar type Rodrigues'formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.

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Section: Closing Remarksmentioning

confidence: 98%

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Durán¹,

Grünbaum²

2005

Journal of Approximation Theory

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“…The study of this question starts with [7]. A first attempt to go beyond the issue of the existence of one nontrivial element in D(W ) and to study (experimentally, with assistance from symbolic computation) the full algebra is undertaken in [5]. An analytical proof of some of the results conjectured in [5] will appear in [32].…”

Section: Grünbaum and Tirao Ieotmentioning

confidence: 99%

The Algebra of Differential Operators Associated to a Weight Matrix

Grünbaum

Tirao

2007

Integr. equ. oper. theory

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Given a weight matrix W (x) of size N on the real line one constructs a sequence of matrix valued orthogonal polynomials, {Pn} n≥0 . We study the algebra D(W ) of differential operators D with matrix coefficients such that PnD = ΛnPn, with Λn in the algebra A of N ×N complex matrices. We study certain representations of this algebra, prove that it is a *-algebra and give a precise description of its isomorphic image inside the algebra A N 0 . Mathematics Subject Classification (2000). Primary 33C45, 47L80; Secondary 47E05.

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“…If one considers multiplication by x or by y we get a "seven-point" formula. Of course it would be interesting, in the spirit of [3] to consider the full algebra of difference operators going along with the Rahman polynomials.…”

Section: The Bispectral Propertymentioning

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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The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: Five instructive examples

Cited by 41 publications

References 34 publications

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

The Algebra of Differential Operators Associated to a Weight Matrix

The Rahman Polynomials Are Bispectral

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