Approximating the Conway-Maxwell-Poisson normalizing constant

The Conway-Maxwell-Poisson is a two-parameter family of distributions on the nonnegative integers. Its parameters  and  model the intensity and the dispersion, respectively. Its normalizing constant is not always easy to compute, so good approximations are needed along with an assessment of their error. Shmueli, et al. [12] derived an approximation assuming that  is an integer, and gave an estimate of the relative error. Their numerical work showed that their approximation performs well in some parameter ranges but not in others. Our aims are to show that this approximation applies to all real  > 0; to provide correction terms to this approximation; and to give different approximations for very small and very large. We then investigate the error terms numerically to assess our approximations. In parameter ranges for which Shmueli's approximation does poorly we show that our correction terms or alternative approximations give considerable improvement.