Asymptotic expansions for solutions of differential equations having a coalescing turning point and double pole, with an application to Legendre functions

The asymptotic behavior of solutions to the second-order linear differential equation $d^{2}w\mathrm{/}d{z}^2=\{u^{2}f(\alpha ,z)+g(z)\}w$ is analyzed for a large real parameter $u$ and $\alpha \in [0,{\alpha }_0]$, where ${\alpha }_0>0$ is fixed. The independent variable $z$ ranges over a complex domain $Z$ (possibly unbounded) on which $f(\alpha ,z)$ and $g(z)$ are analytic except at $z=0$, where the differential equation has a regular singular point. For $\alpha >0$, the function $f(\alpha ,z)$ has a double pole at $z=0$ and a simple zero in $Z$, and as $\alpha \to 0$ the turning point coalesces with the pole. Bessel function approximations are constructed for large $u$ involving asymptotic expansions that are uniformly valid for $z\in Z$ and $\alpha \in [0,{\alpha }_0]$. The expansion coefficients are generated by simple recursions, and explicit error bounds are obtained that simplify earlier results. As an application, uniform asymptotic expansions are derived for associated Legendre functions of large degree $\nu$, valid for complex $z$ in an unbounded domain and for order $\mu \in [0,\nu (1-\delta )]$, where $\delta >0$ is arbitrary.