Binomial Theorem for Negative Index and Negative Parameter

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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:

\(\displaystyle \dfrac 1 {\left({1 - z}\right)^{n + 1} }\) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \binom {-n - 1} k \left({-z}\right)^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \binom {n + k} n z^k\)

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


Proof


Sources