Rahman polynomials

TL;DR

The paper solves for two families of Rahman polynomials by casting them as explicit left eigenvectors of multivariate discrete-time Markov chains built from convolutions of a multinomial with binomial distributions. Degree-one left eigenpolynomials determine the entire $(n\times n)$ parameter set $\{u_{ij}\}$, yielding left eigenfunctions $P_{\boldsymbol m}(\boldsymbol x;\boldsymbol u)$ that are AKomoto-Gelfand-type terminating hypergeometric functions with a multiplicative spectrum $\mathcal{E}(\boldsymbol m)=\prod_{i=1}^n\lambda_i^{m_i}$. The eigenvalues $\lambda_i$ arise as roots of $n\times n$ characteristic equations, and explicit sum rules link them to the base parameters $\boldsymbol\alpha,\boldsymbol\beta$; exceptional degenerate cases are discussed. The generating function of Mizukawa underpins the left eigenvalue equations, confirming the explicit solvability and connecting the polynomials to multivariate hypergeometric structures with symmetric-group invariance. Overall, the work provides a complete, dynamics-driven construction of solvable multivariate Rahman polynomials and clarifies the relationship between dynamics, hypergeometric parameters, and orthogonality measures.

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 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 $\{u_{i\,j}\}$ 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.