Uniform asymptotic expansions for Gegenbauer polynomials and related functions via differential equations having a simple pole

TL;DR

This paper derives simple, accurate, and uniform asymptotic expansions for Gegenbauer polynomials for large degree n, using differential equations with a simple pole, and provides explicit error bounds.

Contribution

It introduces simplified uniform asymptotic expansions for Gegenbauer polynomials based on differential equations with a simple pole, improving computational accuracy and simplicity.

Findings

Expansions are valid for complex arguments including the real interval [0,1]

Error bounds involve only elementary functions

Results extend symmetry to the left half plane

Abstract

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in the right half plane are located: symmetry extends the results to the left half plane. The approximations are derived from the differential equation satisfied by these polynomials, and other independent solutions are also considered. For large $n$ this equation is characterized by having a simple pole, and expansions valid at this singularity involve Bessel functions and slowly varying coefficient functions. The expansions for these functions are simpler than previous approximations, in particular being computable to a high degree of accuracy. Simple explicit error bounds are derived which only involve elementary functions, and thereby provide a simplification of previous expansions and error bounds associated with differential equations having a large parameter and simple pole.