First Born amplitude for transitions from a circular state to a state of large (l, m)

The use of cylindrical polar coordinates instead of the conventional spherical polar coordinates enables us to derive compact expressions of the first Born amplitude for some selected sets of transitions from an arbitrary initial circular \Psin(i),n(i-1),n(i-1)\ state to a final \Psin(f),l(f),m(f)) state of large (l(f), m(f)). The formulae for \Psin(i),n(i-1),n(i-1)\ --> \Psin(f),n(f-1),n(f-2)\ and \Psin(i),n(i-1),n(i-1)\ --> \Psin(f),n(f-1),n(f-3)\ transitions are expressed in terms of the Jacobi polynomials which serve as suitable starting points for constructing complete solutions over the bound energy levels of hydrogen-like atoms. The formulae for \Psin(i),n(i-1),n(i-1)\ --> \Psin(f),n(f-1),n(f-2)\ and \Psin(i),n(i-1),n(i-1)\ --> \Psin(f),n(f-1),n(f-3)\ transitions are in simple algebraic forms and are directly applicable to all possible values of n(i) and n(f). It emerges that the method can be extended to evaluate the first Born amplitude for many other transitions involving states of large (l, m).