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In his monograph [Classical and quantum orthogonal polynomials in onevariable, Cambridge University Press, 2005 (paperback edition 2009)], Ismailconjectured that certain structure relations involving the Askey-Wilsonoperator characterize proper subsets of the set of all$\mathcal{D}_q$-classical orthogonal polynomials, here to be understood as theAskey-Wilson polynomials and their limit cases. In this paper we give twocharacterization theorems for $\mathcal{D}_q$-semiclassical (and classical)orthogonal polynomials in consonance with the pioneering works by Maroni [Ann.Mat. Pura. Appl. (1987)] and Bonan, Lubinsky, and Nevai [SIAM J. Math. Anal. 18(1987)] for the standard derivative, re-establishing in this context theperfect "symmetry" between the standard derivative and the Askey-Wilsonoperator. As an application, we present a sequence of$\mathcal{D}_q$-semiclassical orthogonal polynomials of class two thatdisproves Ismail's conjectures. Further results are presented for Hahn'soperator.

Epilegomena to the study of semiclassical orthogonal polynomials