  Surprisingly, apart from some special cases, simple asymptotic expansions for
the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for
large degree $\nu$ or large order $\mu$ are not available in the literature.
The main purpose of the present paper is to fill this gap by deriving simple
(inverse) factorial expansions for these functions and provide sharp and
realistic bounds on their error terms. Analogous results for the Ferrers
functions and the closely related Gegenbauer function are also included. In the
cases that $\nu$ is an integer or $2\mu$ is an odd integer, many of these new
expansions terminate and provide finite representations in terms of simple
functions. Most of these representations appear to be new. It is well known
that the hypergeometric series can be regarded as a large-$c$ asymptotic
expansion for the hypergeometric function $F(a,b;c;z)$. We also derive
computable bounds for the remainder term of this expansion. To our best
knowledge, no such estimates have been given in the literature prior to this
paper.
