Binomial Theorem for Negative Index and Negative Parameter

Theorem
Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $z \in \R$ be a real number such that $\left\lvert{z}\right\rvert < 1$.

Then:

where $\dbinom {n + k} n$ denotes a binomial coefficient.

Proof
From General Binomial Theorem: