Ellipsoidal Harmonic of the First Kind

The first solution to Lamé's Differential Equation, denoted for , ..., . They are also called Lamé Functions. 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 denotes by , , , and ,