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

TL;DR

This paper develops uniform asymptotic expansions for solutions of a differential equation with a coalescing turning point and double pole, applying these results to derive asymptotics for Legendre functions of large degree and order.

Contribution

It introduces explicit, uniformly valid asymptotic expansions for solutions near a coalescing turning point and double pole, with simplified error bounds and recursive coefficient generation.

Findings

Derived Bessel function approximations for large parameter u

Obtained explicit error bounds for the asymptotic expansions

Applied results to uniform asymptotics of Legendre functions for large degree and order

Abstract

The asymptotic behavior of solutions to the second-order linear differential equation $d^{2}w/dz^{2}=\{u^{2}f(\alpha,z)+g(z)\}w$ 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.