Negated Upper Index of Binomial Coefficient/Corollary 2

Corollary to Negated Upper Index of Binomial Coefficient

Let $n, m \in \Z$.

Then:

$\dbinom n m = \paren {-1}^{n - m} \dbinom {-\paren {m + 1} } {n - m}$

where $\dbinom n m$ is a binomial coefficient.

Proof

 $\displaystyle \dbinom r k$ $=$ $\displaystyle \paren {-1}^k \dbinom {k - r - 1} k$ Negated Upper Index of Binomial Coefficient $\displaystyle \leadsto \ \$ $\displaystyle \dbinom n {n - m}$ $=$ $\displaystyle \paren {-1}^{n - m} \dbinom {\paren {n - m} - n - 1} {n - m}$ setting $r = n$ and $k = n - m$ $\displaystyle \leadsto \ \$ $\displaystyle \dbinom n m$ $=$ $\displaystyle \paren {-1}^{n - m} \dbinom {-m - 1} {n - m}$ Symmetry Rule for Binomial Coefficients $\displaystyle$ $=$ $\displaystyle \paren {-1}^{n - m} \dbinom {- \paren {m + 1} } {n - m}$

$\blacksquare$