Rahman polynomials

Abstract

Two very closely related Rahman polynomials are constructed explicitly as the left eigenvectors of certain multi-dimensional discrete time Markov chain operators $K_n^{(i)}({\boldsymbol x},{\boldsymbol y};N)$, $i=1,2$. They are convolutions of an $n+1$-nomial distribution $W_n({\boldsymbol x};N)$ and an $n$-tuple of binomial distributions $\prod_{i}W_1(x_i;N)$. The one for the original Rahman polynomials is $K_n^{(1)}({\boldsymbol x},{\boldsymbol y};N) =\sum_{\boldsymbol z}W_n({\boldsymbol x}-{\boldsymbol z};N-\sum_{i}z_i) \prod_{i}W_1(z_i;y_i)$. The closely related one is \ $K_n^{(2)}({\boldsymbol x},{\boldsymbol y};N) =\sum_{\boldsymbol z}W_n({\boldsymbol x}-{\boldsymbol z};N-\sum_{i}y_i) \prod_{i}W_1(z_i;y_i)$. The original Markov chain was introduced and discussed by Hoare, Rahman and Gr\"{u}nbaum as a multivariable version of the known soluble single variable one. The new one is a generalisation of that of Odake and myself. The anticipated solubility of the model gave Rahman polynomials the prospect of the first multivariate hypergeometric function of Aomoto-Gelfand type connected with solvable dynamics. The promise is now realised. The $n^2$ system parameters $\{u_{i\,j}\}$ of the Rahman polynomials are completely determined. These $u_{i\,j}$'s are irrational functions of the original system parameters, the probabilities of the multinomial and binomial distributions.