The Rahman Polynomials Are Bispectral

Abstract

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. Dedicated to the memory of Vadim Kuznetsov I first met Vadim at a meeting in Esterel, Canada back in 1994. I only saw him a few more times after that, in Amsterdam, in Leeds, in Louvain-la-Neuve, and the last time in Leganes. Although we never wrote any papers together we shared a number of common interests. I always found Vadim extremely helpful and willing to share his ideas and his vast knowledge. He was also willing to put up with people whose work was not in the mainstream and he usually would have some useful remark or a suggestion to make. I like to think that he would have enjoyed seeing this paper which builds on some remarkable work of another common friend I met for the first time at the same Esterel meeting. PreliminariesStarting about thirty years ago, M. Hoare and M. Rahman, published a number of papers studying a class of statistical models that were nicely associated with "classical" orthogonal polynomials in one variable. The reader can consult [19,20,21,4].In a very recent paper [22], which I have seen before publication thanks to the kindness of M. Rahman, they take a huge step and consider the case of several variables. I propose to call these statistical models with the name Hoare-Rahman, and to call the multivariable polynomials that arise with the name Rahman polynomials. I am thankful to Prof. Rahman for several inspiring conversations on the contents of [22], in different coffee houses and restaurants in Fremont, California on the last days of 2006.The model of interest here captures very well the idea of a game of chance where the player takes a risk that can never improve his/her winnings but is then given a second opportunity to improve his/her lot. I am convinced that models of this kind should be of great interest in several areas of applied mathematics not only in physics (where they were conceived) but also in biology, population dynamics, evolutionary models, etc. On top of their potential for applications the polynomials discovered by M. Rahman are a piece of beauty. I hope that this

Time and band limiting for matrix valued functions: an integral and a commuting differential operator

“…We follow the same notation as in [GPZ15], but we include some motivation here for benefit of the reader. Let W (x) be a matrix weight function in the open interval (a, b) and let {Q w (x)} w∈N 0 be a sequence of matrix orthonormal polynomials with respect to the weight W (x).…”

“…In [GPZ15] we have dealt with the analog of the first property for the case discussed in [PZ16]. In this paper we address the second one of the properties above in the same situation.…”

“…Previous explorations of a commutativity property similar to the one above in the matrix valued case can be seen in [GPZ15,CG15], dealing with a full matrix and a narrow banded one. Our work here is related to that in [GPZ15] but there are some important differences: instead of studying the commutant of a matrix M , which arises in the problem of time and band limiting, we look for a symmetric differential operators of order two commuting with an integral operator, which is the other operator of interest featuring in the time and band limiting problem. This extends the work started in the previous references and is much closer to the early work at Bell Labs in the scalar case.…”

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“…The Krawtchouk polynomials of Griffiths, originally introduced in [9, 13] and rediscovered by Hoare and Rahman [15], are usually defined as some specialization of the Aomoto-Gel'fand hypergeometric series [14]:…”

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“…The fact that the polynomials P i, j (s, t; N) are orthogonal follows from the unitarity of the representation U(R) and from the fact that the states |i, j〉 N are orthonormal. The relation They provide a two-variable generalization of the onevariable Krawtchouk polynomials which are orthogonal with respect to the binomial distribution [14,11,10,9,4,12].…”