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The \(q\)-version of a theorem of Bochner. (English) Zbl 0865.33012
Summary: R. Askey and J. Wilson [Mem. Am. Math. Soc. 319 (1985; Zbl 0572.33012)] found a family of orthogonal polynomials in the variable \(s(k) =\frac12(k +1/k)\) that satisfy a \(q\)-difference equation of the form \[ a(k)(p_n(s(qk))- p_n(s(k)))+ b(k)(p_n(s(k/q))- p_n(s(k)))=\theta_n p_n(s(k)),\quad n=0,1,\dots . \] 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 [F. A. Grünbaum and L. Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, CRM Proc. Lect. Notes. 9, 143–154 (1996; Zbl 0865.33008)] 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 [Douglas A. Leonard, SIAM J. Math. Anal. 13, 656–663 (1982; Zbl 0495.33006)].

MSC:
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
39A10 Additive difference equations
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