The first solution to Lamé's differential equation, denoted for , ..., . They are also called Lamé functions of the first
kind. The product of two ellipsoidal harmonics of the first kind is a spherical
harmonic. Whittaker and Watson (1990, pp. 536-537) write

(1)

(2)

and give various types of ellipsoidal harmonics and their highest degree terms as

1.

2.

3.

4. .

A Lamé function of degree may be expressed as

(3)

where or 1/2, are real and unequal
to each other and to , , and , and

(4)

Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by , , , and ,