A new basis for eigen modes on the Sphere

The spherical harmonics Y l m form a basis of the vector space V (of dimensions 2l + 1) of the eigen functions of the Laplacian on the sphere, with eigen value λl = −l (l+ 1). Here we show the existence of an other basis Φj for V, where Φj(X) ≡ (X · Nj), the l power of the scalar product of the current point with a specific null vector Nj . This very simple formulation makes calculations very easy, including those involving the spherical harmonics themselves. We give the correspondence formulae between the two bases, and the transformation properties of the new basis under an SO(2) subgroup of SO(3). This formulation allows to derive a new relation for the spherical harmonics, which is a finite version of the integral representation formula. It provides also a new development for the special Legendre functions