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.

$\ds \dbinom r k$

$=$

$\ds \paren {-1}^k \dbinom {k - r - 1} k$

$\ds \leadsto \ \ $

$\ds \dbinom n {n - m}$

$=$

$\ds \paren {-1}^{n - m} \dbinom {\paren {n - m} - n - 1} {n - m}$

setting $r = n$ and $k = n - m$

$\ds \leadsto \ \ $

$\ds \dbinom n m$

$=$

$\ds \paren {-1}^{n - m} \dbinom {-m - 1} {n - m}$

$\ds $

$=$

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

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(19)$