Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$A(s)P_n(z(s+1))+B(s)P_n(z(s))+C(s)P_n(z(s-1))={\lambda }_n{P}_n(z(s)),n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},N$$ where $z(s), A(s), B(s), C(s)$aresomefunctionsofthediscreteargument$s$and$N$ may be either finite or infinite. The irreducibility condition $A(s-1)C(s)\mathrm{\ne }0$ is assumed for all admissible values of $s$. In the finite case we assume that there are $N+1$ distinct grid points $z(s),s=0,1,\mathrm{.}\mathrm{.}\mathrm{.},N$ such that $z(i) \ne z(j), : i \ne j$\mathrm{.}If$N=\infty$weassumethatthegrid$z(s)$ has infinitely many different values for different values of $s$. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e. $\lambda_n \ne \lambda_m, n \ne m$\mathrm{.}Thenweshowthatnecessarily:(i)thegrid$z(s)$ is at most quadratic or q-quadratic in $s$; (ii) corresponding polynomials $P_n(z)$ are at most the Askey-Wilson polynomials corresponding to the grid $z(s)$. This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids.