On approximation of ultraspherical polynomials in the oscillatory region

For k≥2 even, and α≥−(2k+1)∕4, we provide a uniform approximation of the ultraspherical polynomials P_k ^(α,α)(x) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g(x)=cb(x)(1−x²)^(α+1)∕2P_k ^(α,α)(x)=cosB(x)+r(x). Here the constant c=c(k,α) is defined by the normalization of P_k ^(α,α)(x), B(x)=∫₀ ^xb(x)dx, and the functions b(x) and B(x), as well as bounds on the error term r(x), are given by some rather simple elementary functions.