From AMS Glossary
An analytic basis function on the sphere that is commonly used in a spectral model.
A spherical harmonic is defined for each total wavenumber n and zonal wavenumber m as the following function of sine of latitude μ and longitude λ:
where Pm,n is the associated Legendre function defined as
The spherical harmonic basis functions satisfy the orthogonal relationship
and they satisfy the elliptic equation on the sphere: