A characterization of Askey-Wilson polynomials

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.

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

“…We present another treatment of generalized Bochner theorem and develop two structure relations for classical orthogonal polynomials of the quadratic and q-quadratic variable: a first structure relation that we use to characterize Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and subcases, including limiting cases when one or more parameters tend to ∞, as the family of classical orthogonal polynomials of the quadratic and qquadratic variable; a second structure relation involving only the divided-difference operator D x , that generalizes the Wilson operator and the Askey-Wilson operator. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [ [21]) as shown in [17] and that of Costas-Santos and Marcellán (cf. [8]) in the present paper, we have illustrated that polynomials appearing in the Askey scheme and q-Askey scheme [18] can be effectively studied by using only the operator D x .…”

“…Since {p n } ∞ n=0 is orthogonal (cf. [17,18]) both families are orthogonal with respect to the same measure. Therefore P n , n = 1, 2, 3, 4, is up to a multiplicative factor equal to p n which satisfies (4.3) by Theorem 3.6).…”

“…The proof is deduced from Theorem 4.2, Corollary 3.9 and the fact that limiting cases of Askey-Wilson polynomials as one or more parameters a, b, c, d tend to ∞ are orthogonal polynomials families (cf. [13,Remark 3.2]), see also [17].…”

“…Coefficients of the first structure relation (4.2) for the Askey-Wilson polynomials have been given in [17]. As for those of the second structure relation we have the following: Substitute c by cq, d by dq.…”

“…Ismail considered the problem of the first structure relation for Askey-Wilson polynomials in the conjecture [14,Conjecture 24.7.9]. In [17,Corollary 3.3], we completed the conjecture by proving that a sequence of monic orthogonal polynomials satisfies the first structure relation π(x)D 2 q P n (x) = 2 j=−2 a n,n+j P n+j (x), a n,n−2 = 0, x = cos θ,…”