1. In this paper I shall deal with the solutions of the Lamé equation

when n and h are arbitrary complex or real parameters and k is any number in the complex plane cut along the real axis from 1 to ∞ and from −1 to −∞. Since the coefficients of (1) are periodic functions of am(x, k), we conclude ](5), § 19·4] that there is a solution of (1), y0(x), which has a trigonometric expansion of the form

where θ is a certain constant, the characteristic exponent, which depends on h, k and n. Unless θ is an integer, y0(x) and y0(−x) are two distinct solutions of the Lamé equation.

It is easy to obtain the system of recurrence relations

for the coefficients cr. θ is determined, mod 1, by the condition that this system of recurrence relations should have a solution {cr} for which

k′ being the principal value of (1−k2)½