Inequalities for zeros of Jacobi polynomials via Sturm's theorem: Gautschi's conjectures
In 2009, \textit{W. Gautschi} [Numer. Algorithms 50, No. 1, 93--96 (2009; Zbl 1165.33009); ibid. 50, No. 3, 293--296 (2009; Zbl 1168.33001)] conjectured two inequalities for the zeros \(x_{n,k}^{(a,b)}\), \(k=1,2,\ldots,n\) of Jacobi polynomials and especially for \(\theta_{n,k}^{(a,b)}=\arccos x_{n,k}^{(a,b)}\). The authors establish the conjectures for large domains of the \((a,b)\) plane. Their main tool is Sturm's comparison theorem for the zeros of solutions of the Sturm-Liouville differential equation. Especially they present a direct consequence of Sturm's theorem and they analyze a particular function that appears in the proof of the theorem.
scientific article; zbMATH DE number 6380840
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