A generalised spheroidal wave equation

Abstract 1.1. The differential equation considered in this paper may be written in the form d/dξ { (1-ξ²) dX/dξ } + { λ²ξ² - 2pλξ - n3²/1-ξ² + μ׳ } X = 0. (1) When p = 0 this equation becomes the equation giving the solution of ∇²X —λ²X = 0 in spheroidal co-ordinates. The equation may therefore be called a generalised spheroidal wave equation. In Part I of this paper we shall consider equation (1) when p ≠ 0, and in Part II, which consists of sections 6 to 10, we shall consider the equation with p = 0. The transformation X = (ξ² — l)n3/²f reduces the equation to one with polynomial coefficients (1 - ξ²) d²f/dξ²- 2 (n3 + 1)ξ df/dξ +{ λ²ξ² - 2pλξ + μ׳ - n3 (n3 + 1) } f = 0. (2)