Convexity of a function with discrete parameters

Question

Let $k\in\mathbb N_{>0}$, $n_1,\dots,n_k\in\mathbb N_{>0}$ and $N=\sum_{i=1}^kn_i$. I would like to know if the following function is convex: $$ r_{\alpha,\theta}(Z) = \overbrace{\sum_{i=1}^{n-1}\ln(\alpha+i)}^\beta - \overbrace{\sum_{i=1}^{k-1}\ln(\alpha+i\theta)}^{\gamma(k)} - \overbrace{\sum_{c=1}^k\sum_{i=1}^{n_c-1}\ln(i-\theta)}^{\delta(Z)}. $$ where $\alpha$ and $\theta$ are constants. I am not sure how I can do this. At first, I was thinking about relaxing the discreteness of $k$ and $n_i$ but it just doesn't make sense.