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 ) ) = λ 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 ) ≠ 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 ) ≠ z ( j ) , i ≠ j . If N = ∞ 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., λ n ≠ λ m , n ≠ 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.