Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to Legendre functions

Linear second-order ordinary differential equations of the form $d^{2}w/dz^{2}=\{u^{2}f(a,z)$ $+g(z)\}w$ are studied for large values of the real parameter $u$, where $z$ ranges over a bounded or unbounded complex domain $Z$, and $a_{0} \le a \le a_{1} < \infty$. The functions $f(a,z)$ and $g(z)$ are analytic in the interior of $Z$. Moreover, $f(a,z)$ has exactly two real simple zeros in $Z$ for $a>a_{0}$ that depend continuously on $a$ and coalesce into a double zero as $a \to a_{0}$. Uniform asymptotic expansions are obtained for solutions in terms of parabolic cylinder functions and their derivatives, together with slowly varying coefficient functions. The coefficients are readily computable and explicit error bounds are provided. The results are then applied to derive new asymptotic expansions for the associated Legendre functions when both the degree $\nu$ and the order $\mu$ are large.