Error bounds for a uniform asymptotic expansion of the Legendre function $P_n^{-m}(\textrm {cosh}\ z)$

For fixed $m$ with $m + \frac {1}{2} > 0$, an asymptotic expansion for large $n$ is derived for the Legendre function $P_n^{ - m}\left ( {\cosh z} \right )$,which is uniformly valid for $z$ in the unbounded interval $\left [ {0, \infty } \right )$. Our method is based on an integral representation of this function. The coefficients in the expansion satisfy a recurrence relation. Simple computable bounds are also constructed for the error terms associated with the expansion.