I've derived a couple of probability distributions that involve the summations: $$ \sum_{k=0}^{n-1}\binom{n+k-1}{k}p^nq^k $$ $$ \sum_{k=0}^{n-1}(n+k)\binom{n+k-1}{k}p^nq^k $$ where $q=1-p$.
I am struggling to derive closed-form expressions for these summations, which I feel ought to be possible given their similarity to the negative binomial distribution. I have tried a variety of approaches, the latest being to write the sum as $S_n$ and consider $qS_n-S_n$, but no joy so far.
Any help is much appreciated!
Hint
If you replace the binomial coefficient by the gamma function $$S_n=\sum_{k=0}^{n-1}\binom{n+k-1}{k}\,p^n\,q^k=\frac{p^n}{\Gamma (n)}\sum_{k=0}^{n-1} \frac{ \Gamma (k+n)}{\Gamma (k+1)}\,q^k$$ and, as usual, the result of the summation is an hypergeometric function.
If $q=(1-p)$ $$S_n=1-\binom{2 n-1}{n} \Big((1-p)\, p\Big)^n \, _2F_1(1,2 n;n+1;1-p)$$ where appears the Gaussian hypergeometric function.