  In his monograph [Classical and quantum orthogonal polynomials in one
variable, Cambridge University Press, 2005 (paperback edition 2009)], Ismail
conjectured that certain structure relations involving the Askey-Wilson
operator characterize proper subsets of the set of all
$\mathcal{D}_q$-classical orthogonal polynomials, here to be understood as the
Askey-Wilson polynomials and their limit cases. In this paper we give two
characterization 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 the
perfect "symmetry" between the standard derivative and the Askey-Wilson
operator. As an application, we present a sequence of
$\mathcal{D}_q$-semiclassical orthogonal polynomials of class two that
disproves Ismail's conjectures. Further results are presented for Hahn's
operator.
