22,854 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
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