The q-version of a theorem of Bochner

Abstract

Askey and Wilson (1985) found a family of orthogonal polynomials in the variable s(k) = 1 2 (k + 1 k ) that satisfy a q-difference equation of the form a(k)(p n(s(qk)) − p n(s(k))) + b(k)(p n(s( k q )) − p n(s(k))) = θ np n(s(k)) , n = 0, 1, …. We show here that this property characterizes the Askey-Wilson polynomials. The proof is based on an “operator identity” of independent interest. This identity can be adapted to prove other characterization results. Indeed it was used in (Grünbaum and Haine, 1996) to give a new derivation of the result of Bochner alluded to in the title of this paper. We give the appropriate identity for the case of difference equations (leading to the Wilson polynomials), but pursue the consequences only in the case of q-difference equations leading to the Askey-Wilson and big q-Jacobi polynomials. This approach also works in the discrete case and should yield the results in (Leonard, 1982).