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The bounds for the error term of an asymptotic approximation of Jacobi polynomials. (English) Zbl 0643.41022
Orthogonal polynomials and their applications, Proc. Int. Symp., Segovia/Spain 1986, Lect. Notes Math. 1329, 203-221 (1988).
[For the entire collection see Zbl 0638.00018.] We consider a new asymptotic approximation of Jacobi polynomials $P\sb n\sp{(\alpha,\beta)}(\cos \theta)$ and we obtain a realistic and explicit bound for the corresponding error term. The approximation is of Hilb’s type and is uniformly valid for $0<\theta \le \pi -\epsilon$, $\epsilon >0$. Bounds for the error term in the asymptotic approximation of the zeros of $P\sb n\sp{(\alpha,\beta)}(\cos \theta)$ are also given.

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A10Approximation by polynomials