In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum appearing in the paper I am preparing:

$$sum_{n=1}^N frac{Gamma(Ln)}{Gamma(Ln+r)},$$

where $L,N$ are positive integers greater than 1 and $r$ is a non-integer with $1<r<2$.

The natural conversion of this sum into an integral, as pointed out in my question, doesn’t seem to help at all, since I can’t compute it.

I was told that the sum is related to the Foxâ€“Wright function. That is useful in the sense that at least I can give a name to my expression, but of course doesn’t help to compute the sum, so I tried to write it as

$$sum_{n=1}^N frac{1}{(Ln)_{r}},$$

which is exactly what appears in http://specialfunctionswiki.org/index.php/Sum_of_reciprocal_Pochhammer_symbols_of_a_fixed_exponent with $L=1$. Does somebody know about a generalization of this result?

Anyway, it would suffice for me to compute the infinite sum, that is $$sum_{n=1}^infty frac{Gamma(Ln)}{Gamma(Ln+r)},$$

which numerically I see that converges, and to know just the asymptotic expansion of a very similar expression,

$$sum_{n=1}^N frac{Gamma(Ln+1)}{Gamma(Ln+r)}$$

as $Ntoinfty$. This last sum doesn’t converge, but I know numerically that its limit when we add some other functions of $N$ exists, so having an asymptotic expansion would be enough.

Thank you so much.