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