Positivity of Turán determinants for orthogonal polynomials II

Abstract

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n\left(1\right)=1,$ satisfy Turán’s inequality if $p_n^2\left(x\right)-p_{n-1}\left(x\right)p_{n+1}\left(x\right)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán’s inequality. This extends essentially the results of Szwarc(1998). In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

Introduction

Consider a symmetric probability measure $\mu$ such that $\mathrm{supp}\mu =[-1,1]$. By the Gram–Schmidt orthogonalization procedure applied to the system of monomials $x^n$, $n\ge 0$, we obtain a sequence of orthogonal polynomials $p_n\left(x\right)$, $n\ge 0$. Every polynomial $p_n$ is of exact degree $n$. We may assume that its leading coefficient is positive. It is well known that the polynomials $p_n$ satisfy the three term recurrence relation of the form

$$x{p}_n={\gamma }_n{p}_{n+1}+{\alpha }_n{p}_{n-1},n\ge 0,$$

with convention ${\alpha }_0=p_{-1}=0$. Due to orthogonality the polynomial $p_n$ has $n$ roots in the open interval $(-1,1)$. Therefore $p_n\left(1\right)>0$. Let

$$P_n\left(x\right)=\frac{p_n\left(x\right)}{p_n\left(1\right)},n\ge 0.$$

The coefficients ${\gamma }_n,{\alpha }_{n+1}$ are positive for $n\ge 0$. In case the polynomials $p_n$ are orthonormal then the sequences of the coefficients are related by ${\gamma }_n={\alpha }_{n+1}$ and the recurrence relation simplifies to

$$x{p}_n={\alpha }_{n+1}{p}_{n+1}+{\alpha }_n{p}_{n-1},n\ge 0.$$

We refer to [5], [16] for the basic theory concerning orthogonal polynomials.

We are interested in determining when

$${\Delta }_n\left(x\right)≔P_n{\left(x\right)}^2-P_{n-1}\left(x\right)P_{n+1}\left(x\right)\ge 0,n\ge 1.$$

The expression ${\Delta }_n\left(x\right)$ is called the Turán’s determinant. The problem has been studied for many classes of specific orthogonal polynomials (see [1], [2], [3], [4], [6], [7], [8], [10], [12], [13], [14], [15], [20], [21]). We refer to the introduction in [18] for a short account of known results.

Turán determinants can be used to determine the orthogonality measure $\mu$ in terms of orthonormal polynomials $p_n$. Paul Nevai [11] observed if ${\alpha }_n\stackrel{n}{\to }1/2$ then the sequence of measures (perhaps signed)

$$[p_n^2\left(x\right)-p_{n-1}\left(x\right)p_{n+1}\left(x\right)]d\mu \left(x\right)$$

is weakly convergent to the measure

$$\frac{2}{\pi }\sqrt{1-x^2}dx,\left|x\right|<1.$$

Maté and Nevai [9] showed that if additionally sequence ${\alpha }_n$ has bounded variation then the limit of Turán determinants exists. Moreover the orthogonality measure is absolutely continuous on the interval $(-1,1)$ its density is given by

$$\frac{2\sqrt{1-x^2}}{\pi f\left(x\right)},\left|x\right|<1,$$

where

$$f\left(x\right)≔\underset{n}{lim}[p_n^2\left(x\right)-p_{n-1}\left(x\right)p_{n+1}\left(x\right)]>0,\left|x\right|<1.$$

It turns out that the way we normalize the polynomials is essential for the Turán inequality to hold. Indeed, assume $p_n$ satisfy (1) and $p_n\left(1\right)=1$, i.e. 

$${\alpha }_n+{\gamma }_n=1.$$

Assume

$$p_n^2\left(x\right)-p_{n-1}\left(x\right)p_{n+1}\left(x\right)\ge 0,\left|x\right|\le 1,n\ge 1.$$

Define new polynomials by $p_n^{\left(\sigma \right)}\left(x\right)={\sigma }_n{p}_n\left(x\right)$, where ${\sigma }_n$ is a sequence of positive coefficients. Then the condition

$${\left\{p_n^{\left(\sigma \right)}\left(x\right)\right\}}^2-p_{n-1}^{\left(\sigma \right)}\left(x\right)p_{n+1}^{\left(\sigma \right)}\left(x\right)\ge 0,\left|x\right|\le 1,n\ge 1$$

is equivalent to (see Proposition [18])

$${\sigma }_n^2-{\sigma }_{n-1}{\sigma }_{n+1}\ge 0,n\ge 1.$$

This means if the Turán determinants are nonnegative, when the polynomials are normalized at $x=1$, then they stay nonnegative for any other normalization provided that they are nonnegative at $x=1$, as ${\sigma }_n=p_n^{\left(\sigma \right)}\left(1\right)$.

By Theorem 1 [18] if the polynomials are normalized at $x=1$, i.e. $p_n\left(1\right)=1$, ${\alpha }_n$ is increasing and ${\alpha }_n\le \frac{1}{2}$, the Turán determinants are positive in the interval $(-1,1)$. This result can be applied to many classes of orthogonal polynomials, including for example the ultraspherical polynomials for which positivity has been obtained in [12], [13]

The result mentioned above can be applied provided that we are given the coefficients ${\alpha }_n$ explicitly. For many classes of orthogonal polynomials in the interval $[-1,1]$ we are given recurrence relations, but the values $p_n\left(1\right)$ cannot be evaluated in the explicit form. Therefore we are unable to provide a recurrence relation for the polynomials $P_n\left(x\right)=p_n\left(x\right)/p_n\left(1\right)$, in the form for which we can inspect easily the assumptions of Theorem 1 [18]. This occurs when we study the associated polynomials. Indeed assume $p_n$ satisfy (1), (3). For a fixed natural number the associated polynomials $p_n^{\left(k\right)}$ of order $k$ are defined by

$$x{p}_n^{\left(k\right)}=\left\{\begin{array}{cc}{\gamma }_k{p}_1^{\left(k\right)}\hfill & n=0,\hfill \\ {\gamma }_{n+k}{p}_{n+1}^{\left(k\right)}+{\alpha }_{n+k}{p}_{n-1}^{\left(k\right)}\hfill & n\ge 1.\hfill \end{array}\right.$$

These polynomials do not satisfy $p_n^{\left(k\right)}\left(1\right)=1$ as

$$p_1^{\left(k\right)}\left(1\right)={\gamma }_k^{-1}={(1-{\alpha }_k)}^{-1}>1.$$

The obstacle described above has been partially overcome in Corollary 1 of [18], but it required additional assumptions, in particular ${\gamma }_0\ge 1$. Unfortunately many examples including the associated polynomials violate that condition. The aim of this note is to provide a counterpart to Corollary 1 [18] by allowing ${\gamma }_0<1$. This is done in Theorem 1. As the assumptions in this theorem are complicated Corollary 1 provides a wide class of relatively simple recurrence relations for which Theorem 1 applies. General examples are provided at the end of the paper.