# 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 $\size z < 1$.

Then:

 $\displaystyle \dfrac 1 {\paren {1 - z}^{n + 1} }$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \binom {-n - 1} k \paren {-z}^k$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \binom {n + k} n z^k$

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

## Proof

 $\displaystyle \dfrac 1 {\paren {1 - z}^{n + 1} }$ $=$ $\displaystyle \paren {1 + \paren {-z} }^{- n - 1}$ $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \binom {- n - 1} k \paren {-z}^k$ General Binomial Theorem $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \dbinom {n + 1 + k - 1} k \paren {-1}^k \paren {-z}^k$ Negated Upper Index of Binomial Coefficient $\displaystyle$ $=$ $\displaystyle \sum_{k \mathop \ge 0} \dbinom {n + k} k z^k$

$\blacksquare$