Approximating the Conway–Maxwell–Poisson distribution normalization constant

By adding a second parameter, Conway and Maxwell created a new distribution for situations where data deviate from the standard Poisson distribution. This new distribution contains a normalization constant expressed as an infinite sum whose summation has no known closed-form expression. Shmueli et al. produced an approximation for this sum but proved only that it was valid for integer values of the second parameter, although they conjectured it was also valid for non-integers. Here we prove their conjecture to be true and discuss for what range of parameters the approximation can be accurately applied.