Using the functional approach, we state and prove a characterization theorem for classical orthogonal polynomials on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable) including the Askey-Wilson polynomials. This theorem proves the equivalence between seven characterization properties, namely the Pearson equation for the linear functional, the second-order divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations, and the Riccati equation for the formal Stieltjes function.

No abstract

We show that the only monic orthogonal polynomials {Pn} ∞ n=0 that satisfyan,n+jPn+j (x), x = cos θ, an,n−2 = 0, n = 2, 3, . . . ,where π(x) is a polynomial of degree at most 4 and Dq is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.

No abstract

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {P n } ∞ n=0 , the orthogonality of the second derivatives {D 2x P n } ∞ n=2 and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to ∞. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.