Epilegomena to the study of semiclassical orthogonal polynomials

In his monograph [M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications (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 Dq-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 Dq-semiclassical (and classical) orthogonal polynomials in consonance with the pioneering works by Maroni [Ann. Mat. Pura. Appl. 149, 165–184 (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 Dq-semiclassical orthogonal polynomials of class two that disproves Ismail’s conjectures. Further results are presented for Hahn’s operator, which in turn is related with another conjecture posed by Ismail.