Can this binomial summation be simplified?

Question

I got something like

$\displaystyle\sum_{i=0}^K{ \binom{n+i}{i} \cdot \alpha^i} $

where $n,\ K,\ \alpha$ are definite values, $\binom{n+i}{i}$ is the Combinatorial number that choose $i$ from $n+i$, can this summation be simplified? Thank you.

Answer 1

The summand, being a product of a binomial coefficient and something to the power of the iterand, looks like a hypergeometric function.

In fact, symbolic algebra reveals that it is equal to $$ \left(1-\alpha\right)^{-n-1}-\alpha^{K+1}\binom{K+n+1}{K+1}{}_{2}F_{1}\left(1,K+n+2;K+2;\alpha\right). $$ Naturally, the trouble term is the hypergeometric $_{2}F_{1}$. I do not think this is a particularly useful form, other than being "closed" in some sense.