Forgive my ignorance here. I am going through "Latent Dirichlet Allocation" (pdf) where they state the following (just assuming common knowledge about the Dirichlet distribution):
$\frac{1}{\delta\alpha_i} log \{ \Gamma (\sum_{j=1}^k \alpha_j) \} = \Psi (\sum_{j=1}^k \alpha_j) = \Psi (\alpha_i + k)$
Here I collect all $a_j$ different from $a_i$ in $k$.
It is easy for me to find out $\frac{\Gamma'(x+k)}{\Gamma(x+k)}$ with $k$ integer, namely by differentiating $\Gamma(x+1)$, realizing it is equal $x \Gamma(x)$, calculating the ratio $\Gamma'/\Gamma$ and doing this $k$ times:
$\frac{\Gamma'(x+k)}{\Gamma(x+k)} = \frac{\Gamma'(x)}{\Gamma(x)} + \sum_{j=0}^{k-1} \frac{1}{x+j}$
And similarly the following is not that hard either for $k$ integer:
$\Psi(x+k)-\Psi(x) = \sum_{j=0}^{k-1} \frac{1}{x+j}$
And because we know $\frac{\Gamma'(x)}{\Gamma(x)} = \Psi(x)$ this allows us to write $\frac{\Gamma'(x+k)}{\Gamma(x+k)} = \Psi(x+k)$.
However, how does this work for positive, real $k$?
