An accurate quantum expression for radiative transition between nearby Rydberg states

The paper gives three main results as follows. (1) An accurate quantum expression of the radial matrix element for radiative dipole transition between nearby Rydberg states. Its remarkable numerical accuracy is demonstrated over a very wide range of principal and orbital angular momentum quantum numbers covering low-lying states to very high Rydberg states. (2) A simple but accurate approximation to a class of terminating hypergeometric functions whose three arguments are large. This result essentially solves the problem of extracting analytic properties and performing numerical computations of such hypergeometric functions over a large range of values of the arguments which were earlier regarded to pose difficulties. (3) A derivation of the formula of the radial dipole matrix element of the correspondence principle method starting from the corresponding quantum expression, which, to the best of our knowledge, was not previously available in the literature