Complex asymptotics in\ lambda for the Gegenbauer functions C_\lambda^\alpha(z) and D_\lambda^\alpha(z) with z\in(-1,1)

Abstract

We derive asymptotic results for the Gegenbauer functions C_\lambda^\alpha(z) and D_\lambda^\alpha(z) of the first and second kind for complex z and the degree \lambda -> \infty, apply the results to the case z \in (-1,1), and establish the connection of these results to asymptotic Bessel-function approximations of the functions for z\rightarrow \pm 1.