  The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space
${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the
Laplacian on the sphere, with eigenvalue $\lambda_{\ell} = -\ell ~(\ell +1)$.
Here we show the existence of a different basis $\Phi ^{\ell}_j$ for ${\cal V}
^{\ell}$, where $\Phi ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell
^{th}$ power of the scalar product of the current point with a specific null
vector $N_j$. We give explicitly the transformation properties between the two
bases. The simplicity of calculations in the new basis allows easy
manipulations of the harmonic functions. In particular, we express the
transformation rules for the new basis, under any isometry of the sphere.
  The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and
back) allows to derive new properties for the $Y_{\ell m}$. In particular, this
leads to a new relation for the $Y_{\ell m}$, which is a finite version of the
well known integral representation formula. It provides also new development
formulae for the Legendre polynomials and for the special Legendre functions.
