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Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions. (English) Zbl 0867.42011
This paper is concerned with the Stieltjes polynomials \(E^{(\lambda)}_{n+1}(x)\) defined by \[ \int^1_{-1}w_\lambda(x)P^{(\lambda)}_n(x)E^{(\lambda)}_{n+1}(x)x^kdx=0,\quad k=0,1,2,\dots,n, \] where \(w_\lambda(x)=(1-x^2)^{\lambda-1/2}\), \(\lambda>-1/2\), is the ultraspherical weight function and \(P^{(\lambda)}_n(x)\) are the ultraspherical polynomials. The author gives an asymptotic representation of \(E^{(\lambda)}_{n+1}(\cos\vartheta)\), as \(n\to\infty\), in the case \(1<\lambda\leq 2\). This representation, which holds uniformly for \(\varepsilon\leq\vartheta\leq\pi-\varepsilon\), \(\varepsilon>0\), completes the study of the ultraspherical weight functions for which Stieltjes polynomials are known to have only real distinct zeros inside \((-1,1)\) for all \(n\). The asymptotic behaviour obtained is successfully applied in proving interesting positivity results for Kronrod extensions of Gauss and Lobatto quadrature rules.

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C55 Spherical harmonics
41A55 Approximate quadratures
Full Text: DOI
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