The Rahman Polynomials Are Bispectral

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.

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/

References in corpus (3)

• Infinite J-matrices and a matrix moment problem
• A Probablistic Origin for a New Class of Bivariate Polynomials
• Random walks and orthogonal polynomials: some challenges

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• A Lie theoretic interpretation of multivariate hypergeometric polynomials
• On a Family of 2-Variable Orthogonal Krawtchouk Polynomials
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• A superintegrable finite oscillator in two dimensions with SU(2) symmetry
• Spin lattices, state transfer and bivariate Krawtchouk polynomials
• The Darboux process and a noncommutative bispectral problem
• Rahman polynomials
• Factorized $A_2$-Leonard pair