Basis sets and Coulomb resolutions in rotational coordinates

Using generalised Laplacian symmetry operators, we construct basis sets or Coulomb resolutions in several separable coordinate systems, including two R-separable systems. This expands the possible geometries in which basis set construction is feasible, a problem which is relevant to both galactic dynamics and computational chemistry. In particular we derive three basis sets (two in prolate spheroidal and one in cylindrical coordinates) which are expressible in closed-form using a single Jacobi polynomial. We also show how any spherical polar or prolate spheroidal basis set may be transformed into a bispherical or toroidal basis set.