The Rahman Polynomials Are Bispectral

TL;DR

This paper demonstrates that Rahman polynomials, recently introduced as eigenfunctions of a Markov chain transition matrix, possess the bispectral property, revealing a significant mathematical characteristic of these polynomials.

Contribution

The paper establishes the bispectral nature of Rahman polynomials, highlighting a new and remarkable property of these recently introduced two-variable polynomials.

Findings

Rahman polynomials are bispectral.

They are eigenfunctions of a Markov chain transition matrix.

The paper suggests further challenges for proof and exploration.

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.