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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. (English) Zbl 1045.33009
There are now several ways to derive an asymptotic expansion for the Jacobi polynomials ${P}_{n}^{\left(\alpha ,\beta \right)}\left(cos\theta \right)$, as $n\to \infty$, which holds uniformly for $\theta \in \left[0,\frac{1}{2}\pi \right]$. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for $\theta$ near the origin, when one takes into account the behavior of the coefficients near $\theta =0$. The aim is to use an alternative method to estimate the remainder. First it is shown that the coefficients in the expansion are bounded for $\theta \in \left[0,\frac{1}{2}\pi \right]$. Next, is given an estimate for the error term which is of the same order as the first neglected term.
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A30 Approximation by other special function classes