The Specific Theories

The preceding chapter has described the general properties of the spaces Y _n (q), but there is a need to fill the general frame with concrete and explicit data. We have seen that the Legendre polynomials contain all that is necessary to determine an orthonormal system of spherical harmonics. This chapter is therefore devoted to the P_n(q; ·) and their many relations and possibilities. This part contains only a selection of identities and special results. An explicit orthogonal basis of y_n (q) was found by Laplace for q = 3. His discovery can be easily extended to higher dimensions. We add a description of the isotropically invariant associated spaces.