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 ofthe 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.
.
may be expressed as | (3) |
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 
,
See also Ellipsoidal Harmonic of the Second Kind, Stieltjes' Theorem
References
Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, pp. 254-258, 1959. Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
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