I was wondering if there is some closed formula for $$\sum_{n=1}^N \frac{\Gamma(Ln)}{\Gamma(Ln+r)},$$ where $L,N$ are positive integers greater than 1 and $r$ is a non-integer with $1<r<2$.
My approach was to rewrite the expression as $$\sum_{n=1}^N \frac{\Gamma(Ln)}{\Gamma(Ln+r)}=\frac{1}{\Gamma(r)}\sum_{n=1}^N\mathrm{B}(Ln,r)$$ and then use the definition $$\mathrm{B}(Ln,r)=\int_0^1 t^{Ln-1}(1-t)^{r-1} dt$$ to put the sum inside the integral. The problem I find is that I don't recognize the resulting integral, which is $$\int_0^1 (1-t)^{r-1}\frac{t^L-t^{L(N+1)}}{t(1-t^L)}dt.$$
Something tells me that a hypergeometric function could appear, but I don't know how to get there.
Thanks in advance!