  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 are located. 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.

Classical Analysis and ODEs