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 $K_n^{(i)}(\mathbf{x},\mathbf{y};N)$, $i=1,2$. They are convolutions of an $n+1$-nomial distribution $W_n(\mathbf{x};N)$ and an $n$-tuple of binomial distributions ${\prod }_i{W}_1(x_i;N)$. The one for the original Rahman polynomials is $K_n^{(1)}(\mathbf{x},\mathbf{y};N)={\sum }_{\mathbf{z}}{W}_n(\mathbf{x}-\mathbf{z};N-{\sum }_i{z}_i){\prod }_i{W}_1(z_i;y_i)$. The closely related one is \ $K_n^{(2)}(\mathbf{x},\mathbf{y};N)={\sum }_{\mathbf{z}}{W}_n(\mathbf{x}-\mathbf{z};N-{\sum }_i{y}_i){\prod }_i{W}_1(z_i;y_i)$. The original Markov chain was introduced and discussed by Hoare, Rahman and Grünbaum 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 $\{u_{ij}\}$ of the Rahman polynomials are completely determined. These $u_{ij}$'s are irrational functions of the original system parameters, the probabilities of the multinomial and binomial distributions.

arXiv commentLaTex2e 21 pages, no figure. Typos are corrected. The construction of another type of Rahman polynomials is added

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