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.
