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Hurwitz's matrices, Cayley transformation and the Cartan-Weyl basis for the orthogonal groups

We find the transformations from the basis of the hydrogen atom of n-dimensions to the basis of the harmonic oscillator of N=2(n-1) dimensions using the Cayley transformation and the Hurwitz matrices. We prove that the eigenfunctions of the Laplacian ∆n are also eigenfunctions of the Laplacien ∆N for n=1, 3, 5 and 9. A new parameterization of the transformation R8->R5 is derived. This research leads us first to a new class of spherical functions of the classical groups we call it the bispherical harmonic functions. Secondly: the development of Hurwitz's matrix in terms of adjoint representation of the Cartan-Weyl basis for the orthogonal groups SO(n) leads to what we call the generating matrices of the Cartan-Weyl basis and then we establish it for n=2,4,8,... .