DETERMINANT INEQUALITIES FOR SIEVED ULTRASPHERICAL POLYNOMIALS

Paul Turan first observed that the Legendre polynomials satisfy the inequality P2 n(x)−Pn−1(x)Pn(x) > 0, −1 < x < 1. Inequalities of this type have since been proved for both classical and nonclassical orthogonal polynomials. In this paper, we prove such an inequality for sieved orthogonal polynomials of the second kind. 2000 Mathematics Subject Classification. Primary 33C47; Secondary 33E30.


Introduction. It was observed by Paul Turán
that the Legendre polynomials satisfy the determinant inequality G. Szegö [8] gave two very beautiful proofs of Turán's inequality.In the years since Szegö's paper appeared, it has been proved by various authors [5,6,7] that inequality (1.1) is satisfied by the classical orthogonal polynomials.In general, let {P n (x)} be a sequence of polynomials orthogonal in an interval [a, b].Then the polynomials must satisfy a recursion where we define P −1 (x) = 0. We begin with a very simple result shows that, inequalities of Turán type are satisfied by any sequence of orthogonal polynomials.
Theorem 1.1.If the polynomials {P n (x)} are orthogonal on a ≤ x ≤ b, then for each n there exists c n , a ≤ c n ≤ b, such that Proof.Consider the quotient Obviously, the roots of P n (x) are singularities and apart from these points f n (x) is continuous.Also, for each root x k,n of P n (x), k = 1, 2,...,n, there is an open interval I n centered at x kn in which P n+1 (x)P n−1 (x) < 0. This follows from the recursion (1.2).
Consequently, f n (x) is bounded above and must take on a positive maximum value at a point c n ∈ [a, b].Thus This proves the theorem.
Although Theorem 1.1 asserts that Turán's inequality is a simple consequence of orthogonality, it is generally quite difficult to determine the point c n referred to in Theorem 1.1.In the case of the classical orthogonal polynomials of Jacobi and their special case, the Gegenbauer polynomials, the point c n = 1 is an endpoint of the interval of orthogonality [−1, 1].Turán's inequality for these classical polynomials is established by using differential identities that are characteristic of classical polynomials.This seems to be the only case that lends itself to that technique.
Obviously, if Inequalities of the form (1.7) will be called weak Turán inequalities to distinguish them from (1.3) which will simply be referred to as Turán inequalities.

A weak Turán inequality for sieved ultraspherical polynomials of the second kind.
The sieved ultraspherical polynomials were discovered by Al-Salam et al. [1].Ismail [2,4] investigated them at great length.If k ≥ 2 is an integer, then the sieved ultraspherical polynomials of the second kind, B λ n (x; k), satisfy where [4] proved the following remarkable formula that is critical in deriving a weak Turán inequality for these polynomials,
Remark 2.2.The factor (m + λ + 1)/(m + 1) that appears in the statement of Theorem 2.1 cannot be improved with a smaller number.This is because of the asymptotic relation (2.16)

3.
A further determinant of sieved ultraspherical polynomials.In [3], Bustoz and Savage proved an inequality for ultraspherical polynomials of the form This inequality was used to answer a conjecture of Askey and Gasper regarding a trigonometric kernel.The question arises if a similar inequality might hold for sieved ultraspherical polynomials.Here, we prove that the analogous inequality does not hold and we determine the location of sign changes for the corresponding sieved expression.We begin with some lemmas.
, where {C λ n (x)} are the ultraspherical polynomials.Then where Proof.Each side of (3.2) is a polynomial of degree 2n + 1.Thus, identity holds in (3.2) if both sides are equal at 2n + 1 points.We prove equality at the roots C λ n (x) = 0, C λ n+1 (x) = 0. First, note that (3.2) holds when x = 0. Thus, we may focus on the nonzero roots of C λ n (x) and C λ n+1 (x).Then it follows that the right-hand side of (3.2) reduces to −C λ n+1 (a)C λ+1 n (a).Thus, (3.2) holds at the n roots of C λ n (x).In a very similar fashion, it can be shown that (3.2) holds at the n + 1 roots of C λ n+1 (x).This proves the lemma.
By iterating (3.2) we get the following corollary.