Turán Inequalities and Zeros of Orthogonal Polynomials

Abstract

We use Turán type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence p k+1 = xp k − c k p k−1 , with a nondecreasing sequence {c k }. As a special case they include a non-asymptotic version of Máté, Nevai and Totik result on the largest zeros of orthogonal polynomials with c k = c k 2δ (1 + o(k −2/3 )). Our proof is based on new Turán inequalities which are obtained by analogy with higher order Laguerre inequalities.

On zeros of discrete orthogonal polynomials

“…Another approach is discussed in [10,11]. For symmetric orthogonal polynomials sharp second order asymptotics and inequalities are known in terms of the three term recurrence relation [8,13]. When applied to the symmetric Hahn and Krawtchouk polynomials they yield bounds very similar, in fact, different only by the constant before the second order term, to those obtained in this paper.…”

Turán inequalities for three-term recurrences with monotonic coefficients

“…This seems to be the only known result of this type for an infinite interval of orthogonality. In [13] we gave some new Turán inequalities for symmetric polynomials and used them to give a non-asymptotic version of Theorem 1, yet with an unavoidably weaker constant in the second-order term, since the bounds obtained hold for any k.…”

“…[20]), it is worth trying to look for higher order generalizations of Turán inequalities. Some results in this direction were given in [5,13] and recently this question was raised again by Nevai [18].…”

“…In terms of the coefficients of the three-term recurrence, very general conditions for the validity of (2) were given in [25], provided that the support of the corresponding measure is finite or half-infinite, such that the polynomials can be normalized to 1 at some point. The symmetric case b i ≡ 0 was considered in [2,13]. Conditions for the validity of Turán type inequalities in a more general setting, for the recurrence y k (x) = a k (x)y k+1 (x) + b k (x)y k−1 (x) in terms of a k (x) and b k (x), were obtained in [23,22].…”

“…for recovering the absolutely continuous part of the corresponding orthogonality measure and for bounding the extreme zeros (see e.g. [1,3,5,13,17]). In terms of the coefficients of the three-term recurrence, very general conditions for the validity of (2) were given in [25], provided that the support of the corresponding measure is finite or half-infinite, such that the polynomials can be normalized to 1 at some point.…”