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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 \(P_ n^{(\alpha,\beta)}(\cos \theta)\) 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 \(0<\theta \leq \pi -\epsilon\), \(\epsilon >0\). Bounds for the error term in the asymptotic approximation of the zeros of \(P_ n^{(\alpha,\beta)}(\cos \theta)\) are also given.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A10 Approximation by polynomials

Citations:

Zbl 0638.00018