## Abstract

New expansions for the Legendre functions $P_{n}^{-m}$(z) and $Q_{n}^{-m}$(z) are obtained; m and n are large positive numbers, 0 < m < n and $\alpha $ = m/(n + $\frac{1}{2}$) is kept fixed as n $\rightarrow \infty $; z is an unrestricted complex variable. Three groups of expansions are obtained. The first is in terms of exponential functions. These expansions are uniformly valid as n $\rightarrow \infty $ with respect to z for all z lying in $\scr{R}$z $\geq $ 0 except for the strips given by $|\scr{I}$z$|$ < $\delta $, $\scr{R}$z < $\beta $ + $\delta $, where $\delta $ > 0 and $\beta $ = $\surd $(1 - $\alpha ^{2}$). The second set of expansions is in terms of Airy functions. These expansions are uniformly valid with respect to z throughout the whole z plane cut from +1 to -$\infty $ except for a pear-shaped domain surrounding the point z = -1 and a strip lying immediately below the real z axis for which $|\scr{R}$z$|$ < $\beta $ + $\delta $, 0 $\geq \scr{I}$z > -$\delta $. The third group of expansions is in terms of Bessel functions of order m. These expansions are valid uniformly with respect to z over the whole cut z plane except for the pear-shaped domain surrounding z = -1. No expansions have been given before for the Legendre functions of large degree and order.