Asymptotics of a Gauss hypergeometric function with large parameters, III: Application to the Legendre functions of large imaginary order and real degree

We obtain the asymptotic expansion for the Gauss hypergeometric function \[F(a-\lambda,b+\lambda;c+i\alpha\lambda;z)\] for $\lambda\rightarrow+\infty$ with $a$, $b$ and $c$ finite parameters by application of the method of steepest descents. The quantity $\alpha$ is real, so that the denominatorial parameter is complex and $z$ is a finite complex variable restricted to lie in the sector $|\arg (1-z)|<\pi$. We concentrate on the particular case $a=0$, $b=c=1$, which is associated with the Legendre functions of real degree and imaginary order. The resulting expansions are of Poincar\'e type and hold in restricted domains of the $z$-plane. An expansion is given at the coalescence of two saddle points. Numerical results illustrating the accuracy of the different expansions are given.