Proof of equality involving logged fraction of Gamma functions Reading up on the log-likelihood of the Zero-Inflated Negative Binomial Regression Model, I have noticed an equality that I have not known before. Since I am not too familiar with Gamma functions involving non-integers, I find myself unable to prove the equality and thus I am looking for help.
The equality goes
$$
\ln \left(\frac{\Gamma(x_i + \theta)}{\Gamma(\theta)}\right) = \sum_{j=0}^{x_i -1}\ln(j+\theta)\text,
$$
where $\theta\in[0,1], x_i \in \mathbb N_0$. See this link, equation D-12 for a similar formulation.
I would appreciate any help on this subject.

$$\sum_{j=0}^{x_i-1} ln(j+\theta )$$
$$=ln(\theta )+ln(1+\theta )+ln(2+\theta ).......ln(x_i-1+\theta)$$
$$=ln(\frac{(\theta -1)!\times\theta.(1+\theta).(2+\theta)...... (x_i-1+\theta)}{(\theta -1)!})$$
$$=ln(\frac{\Gamma(x_i+\theta)}{\Gamma(\theta ) } )
$$
