how to compute $\sum{\frac{(s+k)!}{s!k!}*x^k}$

Question

For $\sum_{k=0}^{\infty}{\frac{(s+k)!}{s!k!}x^k}$, $0\leq x\leq1$. It is not binomial. So how can we simplify the factorial?

Answer 1

It is binomial:

$\frac{(s+k)!}{s!k!} = \frac{1}{k!}(s+k)(s+k-1)\cdots(s+1) = (-1)^k\frac{1}{k!}(-s-1)(-s-2)\cdots (-s-k) = (-1)^k\binom{-s-1}{k}$

so the sum is

$\sum_{k=0}^\infty \binom{-s-1}{k}(-x)^k = (1-x)^{-s-1}$