An infinite set of one-range addition theorems without an infinite second series, for Slater orbitals and it derivatives, applicable more than one coordinate system

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves and Coulomb potentials, and the like, into a sum over Spherical Harmonics that allows the angular integration to be carried out. These have historically been ``two-range'' addition theorems, characterized by the two-fold notation $r_{>}=Max[r_{1},r_{2}]$ and $r_{