By means of a new representation, the Dirac-Coulomb spherical wave functions are treated in a manner which brings out the close formal similarity between these solutions and the spherical wave solutions for the free-electron problem. The radial functions in the new representation have the same form as the non-relativistic radial Coulomb functions, but with an irrational orbital "angular momentum," $l(\ensuremath{\gamma})$. This representation is utilized to deduce a general recursion relation for radial Coulomb eigenfunctions, and show the existence of the Coulomb helicity operator as a constant of the motion. The advantages and properties of this formulation are discussed briefly.