On approximation of ultraspherical polynomials in the oscillatory region

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.