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.

References in corpus (2)

• On zeros of polynomials and allied functions satisfying second order differential equations
• On the asymptotic behavior of Jacobi polynomials with first varying parameter