Bounds on Turán determinants. (English) Zbl 1190.33012
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.
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.
Reviewer: Rodica D. Costin (Columbus)
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
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