[ERG].

This paper collects a large deal of what is presently known about spherical harmonics on the Heisenberg group and the related functions C~a,B). It contains both new results and new approaches to old results. First, orthogonality properties and generating functions for C~a,B) are discussed. Next a new approach to Koranyi's Kelvin transform on the Heisenberg group is given. After a treatment of Heisenberg harmonics, the Kelvin transform is applied in order to obtain a new proof of Dunkl's ex pansion of the translate of the fundamental solution for L • Finally it y is shown that, if the Dirichlet problem for solvable, then the related functions C (a' B) k L on the Heisenberg ball is y form a complete system.