Exact Bound-Bound Gaunt Factor Values for Quantum Levels up to n=2000

Comparison of observations of radio recombination lines in the interstellar medium with theoretical models can be used to constrain electron temperature and density of the gas. An important component of the models is spontaneous transition rates between bound levels. Calculating these rates relies on accurate bound-bound oscillator strengths, which can be cast in terms of the Gaunt factor. The Gaunt factor contains terminating hypergeometric functions that cannot be calculated with sufficient accuracy for high quantum levels ($n \gtrsim 50$) by standard machine-precision methods. Methods to overcome the accuracy problem have been developed, which include asymptotic expansions and recursion relations. These methods, used in astrophysical models to calculate oscillator strengths, can introduce errors, sometimes up to as much as $\sim 8$ per cent. Detections of radio recombination lines with the new Low Frequency Array (LOFAR) has prompted an examination of theoretical models of the interstellar medium. We revisit the calculation of the Gaunt factor, employing modern arbitrary-precision computational methods to tabulate the Gaunt factor for transitions up to quantum level $n=2000$, sufficient to model low frequency Carbon radio recombination lines. The calculations provide a relative error of $\sim3\times 10^{-4}$ when compared to more detailed calculations including relativistic corrections. Our values for the Gaunt factor are provided for download in a tabular format to be used for a wide range of applications.