Negated Upper Index of Binomial Coefficient

Theorem

Let $r \in \R, k \in \Z$.

Then:

$\displaystyle \binom r k = \left({-1}\right)^k \binom {k - r - 1} k$

where $\displaystyle \binom r k$ is a binomial coefficient.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \binom r k\)	\(=\)	\(\displaystyle \frac {r^{\underline k} }{k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of binomial coefficient
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {r \left({r-1}\right) \left({r-2}\right) \cdots \left({r-k+1}\right)}{k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1}\right)^k \frac {\left({- r}\right) \left({- \left({r-1}\right)}\right) \left({- \left({r-2}\right)}\right) \cdots \left({-\left({r-k+1}\right)}\right)}{k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1}\right)^k \frac {\left({k - r - 1}\right) \left({k - r - 2}\right) \left({k-r-3}\right) \cdots \left({k-r-k}\right)}{k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1}\right)^k \frac {\left({k - r - 1}\right) \left({k - r - 2}\right) \left({k-r-3}\right) \cdots \left({k-r-1 - \left({k-1}\right)}\right)}{k!}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({-1}\right)^k \binom {k - r - 1} k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.6 \ \text{G}$