Lamé Polynomials of Large Order

Abstract

Lamé’s equation \[w'' + \left\{ {h - N(N + 1)k^2 sn^2 z} \right\}w(z) = 0\]is an example of a two parameter eigenvalue problem in ordinary differential equations. Here we present new results for the value of h when $Nk$ and $Nk'$ assume large, real values. Uniform asymptotic expansions for Lamé polynomials are also derived. All asymptotic solutions of Lamé’s equation are presented in conjunction with a realistic error bound and constitute a uniform reduction of free variables. The asymptotic expansions are derived for values of z on the rectangle bounded by the lines $\operatorname{Im} (z) = 0$, $K'$, $\operatorname{Re} (z) = 0$, K. Some of the results presented here replace known nonu.niform results whilst others appear to be entirely new.