Bounds on Turán determinants (Q1040859)

The paper considers the polynomials \(p_n\) orthogonal with respect to a symmetric measure \(\mu(x)\) on \([-1,1]\), normalized such that \(p_n(1)=1\). It is shown that the normalized Turán determinant \(\Delta_n(x)/(1-x^2)\) where \(\Delta_n=p_n^2-p_{n-1}p_{n+1}\), is a Turán determinant of order \(n-1\) for orthogonal polynomials with respect to \((1-x^2)\mu(x)\). This property is used to find upper and lower bounds for the normalized Turán determinant on the interval \((-1,1)\) under some monotonicity assumptions on the coefficients of the two step recurrence relation that \(p_n\) satisfies.