48 research outputs found
The Rahman Polynomials Are Bispectral
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.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Some Noncommutative Matrix Algebras Arising in the Bispectral Problem
I revisit the so called "bispectral problem" introduced in a joint paper with
Hans Duistermaat a long time ago, allowing now for the differential operators
to have matrix coefficients and for the eigenfunctions, and one of the
eigenvalues, to be matrix valued too. In the last example we go beyond this and
allow both eigenvalues to be matrix valued
A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application
The one variable Krawtchouk polynomials, a special case of the
function did appear in the spectral representation of the transition kernel for
a Markov chain studied a long time ago by M. Hoare and M. Rahman. A
multivariable extension of this Markov chain was considered in a later paper by
these authors where a certain two variable extension of the Appel
function shows up in the spectral analysis of the corresponding transition
kernel. Independently of any probabilistic consideration a certain
multivariable version of the Gelfand-Aomoto hypergeometric function was
considered in papers by H. Mizukawa and H. Tanaka. These authors and others
such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the
Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the
general -dimensional case. All of these authors proved several properties of
these functions. Here we show that these functions play a crucial role in the
spectral analysis of the transition kernel that comes from pushing the work of
Hoare-Rahman to the multivariable case. The methods employed here to prove this
as well as several properties of these functions are completely different to
those used by the authors mentioned above
Matrix differential equations and scalar polynomials satisfying higher order recursions
We show that any scalar differential operator with a family of polyno- mials
as its common eigenfunctions leads canonically to a matrix differen- tial
operator with the same property. The construction of the correspond- ing family
of matrix valued polynomials has been studied in [D1, D2, DV] but the existence
of a differential operator having them as common eigen- functions had not been
considered This correspondence goes only one way and most matrix valued
situations do not arise in this fashion.
We illustrate this general construction with a few examples. In the case of
some families of scalar valued polynomials introduced in [GH] we take a first
look at the algebra of all matrix differential operators that share these
common eigenfunctions and uncover a number of phenomena that are new to the
matrix valued case