On a conjectured inequality for the largest zero of Jacobi polynomials
Recently [\textit{W. Gautschi} and \textit{P. Leopardi}, Numer. Algorithms 45, No.~1--4, 217--230 (2007; Zbl 1124.33009)], it was conjectured that, given \(\alpha>-1\), \(\beta>-1\), the largest zeros \(x_n^{(\alpha,\beta)}=\cos\theta_n^{(\alpha,\beta)}\), \(n=1,2,\dots\), of the Jacobi polynomials \(P_n^{(\alpha,\beta)}(x)\) satisfy the inequality \(n\theta_n^{(\alpha,\beta)}<(n+1)\theta_{n+1}^{(\alpha,\beta)}\) for all \(n>1\) iff this inequality holds true for \(n=1\). In the present paper, the author shows that the domain in the \((\alpha,\beta)\) plane of validity of the above inequality for all \(n\geq1\) requires some more detailed description. Namely, it is known that the \(n\)-large asymptotics of the Jacobi polynomial \(P_n^{(\alpha,\beta)}(x)\) in some region of the independent variable \(x\) can be described in terms of the Bessel function \(J_{\alpha}\). Taking into account four initial terms in the asymptotic expansion in \(n^{-1}\) of the ratio \(\theta_n^{(\alpha,\beta)}/\theta_{n+1}^{(\alpha,\beta)}\), the author observes a certain domain in the \((\alpha,\beta)\)-plane where the considered inequality is false for all sufficiently large \(n\) even though it holds true for \(n=1\). The paper is concluded by a new conjecture concerning the above inequality.
scientific article; zbMATH DE number 5549491