A characterization of continuous q-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials

A structure relation for some specific orthogonal polynomials

“…Then (P n ) n≥0 satisfies the following other relation (ax − c)U 2 (x)D q P n (x) = r [1] n P n+2 (x) + r [2] n P n+1 (x) + r [3] n P n (x) (2.6) + r [4] n P n−1 (x) + r [5] n P n−2 (x) , for each n = 0, 1, 2, . .…”

“…The system of equations (2.7)-(2.10) is non-linear and so, in general it is not easy to solve. Nevertheless, the same system was solved in [2] for the case where (Pn) n≥0 is a classical orthogonal polynomial and so, it was possible among this known class of orthogonal polynomials to find those which satisfy (2.7)-(2.10). For the present case the task is more harder because we do not have such information and even the initial conditions are different.…”

“…Although (1.2) is a simple relation, this problem was only solved recently due to the complexity of the Askey-Wilson operator and his properties. For instance, it is proved in [1] (for the case π(x) = 1) and in [2] that the only solutions of (1.2) are the some particular cases of the Al-Salam Chihara polynomials, the Chebyshev polynomials of the first kind and the continuous q-Jacobi polynomials. The second part of this conjecture is disproved in [4], where the authors provide a counterexample to (1.3).…”

A counterexample to a conjecture of M. Ismail

“…The previous proof by induction was elaborated after having found the sequence (P n ) n≥0 by a method similar to the one used in [1], which involves a high degree of technical complexity. However, as Galileo reputedly said: "All truths are easy to understand once they are discovered.…”

“…where, for each polynomial f , f (e iθ ) = f (cos θ) and e(x) = x (see [3,Section 12.1]). In [1], we give positive answer to the first part of the following Ismail's conjecture (see [3,Conjecture 24.7.8]):…”