Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 065, 11 pages      arXiv:0705.0468
Contribution to the Vadim Kuznetsov Memorial Issue

The Rahman Polynomials Are Bispectral

F. Alberto Grünbaum
Department of Mathematics, University of California, Berkeley, CA 94720, USA

Received February 01, 2007, in final form April 22, 2007; Published online May 03, 2007

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.

Key words: bispectral property; multivariable polynomials; rings of commuting difference operators.

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