On approximation of ultraspherical polynomials in the oscillatory region

TL;DR

This paper develops a uniform approximation for ultraspherical polynomials in their oscillatory region, providing explicit error bounds and expressing the polynomials in terms of elementary functions, applicable to a broad range of parameters.

Contribution

It offers a new explicit uniform approximation with error bounds for ultraspherical polynomials in the oscillatory region, extending the applicability to all relevant alpha values.

Findings

Explicit error bounds for ultraspherical polynomial approximation

Representation of polynomials using elementary functions and cosine terms

Applicable to all alpha where the oscillatory region is defined

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.