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

TL;DR

This paper derives complex asymptotic formulas for Gegenbauer functions of the first and second kind as their degree grows large, focusing on real arguments within (-1,1) and linking to Bessel function approximations near the endpoints.

Contribution

It provides new asymptotic results for Gegenbauer functions of complex degree, connecting them to Bessel function approximations for real arguments in (-1,1).

Findings

Asymptotic formulas for Gegenbauer functions as degree tends to infinity.

Connection established between Gegenbauer asymptotics and Bessel functions near endpoints.

Results applicable to complex z in the interval (-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.