Generalized Bochner theorem: characterization of the Askey-Wilson polynomials

Abstract

Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$A(s)P_n(z(s+1))+B(s)P_n(z(s))+C(s)P_n(z(s-1))=\hat{I}{»}_n{P}_n(z(s)),n=0,1,2,...,N$$ where $z(s),A(s),B(s),C(s)$ are some functions of the discrete argument $s$ and $N$ may be either finite or infinite. The irreducibility condition $A(s-1)C(s)\ne 0$ is assumed for all admissible values of $s$. In the finite case we assume that there are $N+1$ distinct grid points $z(s),s=0,1,...,N$ such that $z(i)\ne z(j),i\ne j$. If $N=\infty$ we assume that the grid $z(s)$ 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. $\hat{I}{»}_n\ne \hat{I}{»}_m,n\ne m$. Then we show that necessarily: (i) the grid $z(s)$ 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.

16 pages