They occur naturally when applying an orthonormal basis of functions on the unit sphere that transform in a particular way under rotations in three dimensions.
In connection with the quantum theory of atomic structure, John C. Slater defined the integral of three spherical harmonics as a coefficient
Note that the product of two spherical harmonics can be written in terms of these coefficients.
By expanding such a product over a spherical harmonic basis with the same order one may then multiply by
and integrate, using the conjugate property and being careful with phases and normalisations: Hence These coefficient obey a number of identities.