Asymptotic expansions for Legendre functions via differential equations having coalescing turning points

T. M. Dunster

Published 2025-04-28, updated 2025-05-07Version 3

Linear second-order linear ordinary differential equations equations of the form $d^{2}w/dz^{2}=\{u^{2}f(a,z)+g(z)\}w$ are considered for large values of the real parameter $u$, $z$ being a complex variable ranging over a bounded or unbounded complex domain $Z$, and $a_{0} \leq a \leq a_{1} < \infty$. It is assumed that $f(a,z)$ and $g(z)$ are analytic within $Z$. Furthermore, $f(a,z)$ 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 derived for solutions of the equation which involve parabolic cylinder functions and their derivatives, along with certain slowly varying coefficient functions. The new results involve readily computable coefficients and explicit error bounds, and are then applied to provide new asymptotic expansions for the associated Legendre functions when both the degree $\nu$ and order $\mu$ are large.