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)
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(2)
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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)
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where or 1/2, are real and unequal to each other and to , , and , and
(4)
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Byerly (1959) uses the recurrence relations to explicitly compute some ellipsoidal harmonics, which he denoted by , , , and ,
(5)
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(8)
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(9)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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