  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.
