On approximation of ultraspherical polynomials in the oscillatory region

Abstract

For $k \ge 2$ even, and $ \alpha \ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(\alpha,\, \alpha)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all $\alpha$ for which the expression "oscillatory region" makes sense. We show that there the function $g(x)={c \sqrt{b(x)} \, (1-x^2)^{(\alpha+1)/2} P_k^{(\alpha, \alpha)}(x)=\cos \mathcal{B}(x)+ r(x)}$, where $c=c(k, \alpha)$ is defined by the normalization, $\mathcal{B}(x)=\int_{0}^ x b(x) dx$, and the functions $c,\, b(x), \, \mathcal{B}(x)$, as well as bounds on the error term $r(x)$ are given by some rather simple elementary functions.