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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.

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