How can I efficiently calculate the negative binomial cumulative distribution function?

This post is really helpful:

How can I efficiently calculate the binomial cumulative distribution function?

(Title = How can I efficiently calculate the binomial cumulative distribution function?)

However, I need the negative binomial cumulative distribution function.

Is there a way to tweek the code to get a negative cumulative distribution function?

Solution

You can compute the CDF by summing the terms of the PMF taking advantage of the recurrence relationship the terms satisfy. The terms in the series are a little complicated, but the ratio of consecutive terms is simple.