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First, here's a example of the problem I'm trying to solve:

Let's say my data is the total number of times that a patient was visited by a doctor while staying at the hospital.

  • It's count data (doctor visits per day), so I need a discrete distribution.
  • A patient must be visited at least once in order to have a data-point, so the data is zero truncated.
  • The doctor visits per day is overdispersed (I hope I'm using that term correctly) because some days are extra busy at the hospital (fewer visits) and some days are slow (more visits), so I'm choosing a Negative Binomial (instead of Poisson) distribution.

I'm trying to derive a closed form density distribution for a Zero Truncated Negative Binomial subject to an exposure (number of days they were in the hospital).

The PFM of the ZTNB is $$p_{ZTNB}(x) = \frac{Γ(x+r) p^r (1-p)^x}{Γ(n) x! (1-p^r)}, x>0$$

So the density distribution for the exposure model I want is $$p_{exposure}(x) = \frac{x\frac{Γ(x+r) p^r (1-p)^x}{Γ(n) x! (1-p^r)}}{\sum_{x=1}^{\infty} x\frac{Γ(x+r) p^r (1-p)^x}{Γ(n) x! (1-p^r)}}, x>0$$

The numerator is already closed form, so it's just about figuring out if there's a closed form solution for the denominator. Also, I can assume that n=1 so the equation I'm trying to solve for is:

$$\sum_{x=1}^{\infty} x\frac{Γ(x+r) p^r (1-p)^x}{x! (1-p^r)}$$

Any ideas on how/if I can solve this?

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