The Kustaanheimo-Stiefel transformation and certain special functions

The Kustaanheimo-Stiefel transformation is briefly described in various frameworks. This transformation is used to convert the R3 harmonics into R4 harmonics. Then, the Schroedinger equation for an hydrogen-like atom is transformed into the set of a coupled pair of Schroedinger equations for two R2 isotropic harmonic oscillators and a coupled pair of constraint relations. This connection between two famous quantization cases is tackled in terms of both eigenvalues and eigenvectors corresponding to the discrete spectrum of the hydrogen atom. This leads to an integral involving Laguerre, Legendre, and Hermite polynomials. A program has been realized in the algebraic and symbolic programming system macsyma to cover the various computing aspects of this work