  The Kramers-Pasternack relations are used to compute the moments of r (both
positive and negative) for all radial energy eigenfunctions of hydrogenic
atoms. They consist of two algebraic recurrence relations, one for positive
powers and one for negative. Most derivations employ the Feynman-Hellman
theorem or a brute-force integration to determine the second inverse moment,
which is needed to complete the recurrence relations for negative moments. In
this work, we show both how to derive the recurrence relations algebraically
and how to determine the second inverse moment algebraically, which removes the
pedagogical confusion associated with differentiating the Hamiltonian with
respect to the angular momentum quantum number l in order to find the inverse
second moment.
