  The main difficulty in utilizing the O(4) symmetry of the Hydrogen atom in
practical calculations is the dependence of the Fock stereographic projection
on energy. This is due to the fact that the wave functions of the states with
different energies are proportional to the hyperspherical harmonics (HSH)
corresponding to different points on the hypersphere. Thus, the calculation of
the matrix elements reduces to the problem of re-expanding HSH in terms of HSH
depending on different points on the hypersphere. We solve this problem by
applying the technique of multipole expansions for four-dimensional HSH. As a
result, we obtain the multipole expansions whose coefficients are the matrix
elements of the boost operator taken between hydrogen wave functions (i.e.
hydrogen form-factors). The explicit expressions for those coefficients are
derived. It is shown that the hydrogen matrix elements can be presented as
derivatives of an elementary function. Such an operator representation is
convenient for the derivation of recurrency relations connecting matrix
elements between states corresponding to different values of the quantum
numbers $n$ and $l$.
