Negated Upper Index of Binomial Coefficient/Corollary 1

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Corollary to Negated Upper Index of Binomial Coefficient

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

Then:

$\dbinom {-r} k = \paren {-1}^k \dbinom {r + k - 1} k$

where $\dbinom {-r} k$ is a binomial coefficient.


Proof 1

\(\ds \binom {-r} k\) \(=\) \(\ds \paren {-1}^k \binom {k - \paren {-r} - 1} k\) Negated Upper Index of Binomial Coefficient
\(\ds \) \(=\) \(\ds \paren {-1}^k \binom {r + k - 1} k\)

$\blacksquare$


Proof 2

\(\ds \binom {-r} k\) \(=\) \(\ds \frac {\paren {-r}^{\underline k} } {k!}\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \frac {-r \paren {-r - 1} \paren {-r - 2} \dotsm \paren {-r - k + 1} } {k!}\)
\(\ds \) \(=\) \(\ds \paren {-1}^k \frac {\paren r \paren {r + 1} \paren {r + 2} \dotsm \paren {r + k - 1} }{k!}\)
\(\ds \) \(=\) \(\ds \paren {-1}^k \frac {\paren {r + k - 1} \paren {r + k - 2} \paren {r + k - 3} \dotsm \paren {r + k - k} } {k!}\) reversing the order
\(\ds \) \(=\) \(\ds \paren {-1}^k \frac {\paren {r + k - 1} \paren {r + k - 2} \paren {r + k - 3} \dotsm \paren {r + k - 1 - \paren {k - 1} } } {k!}\)
\(\ds \) \(=\) \(\ds \paren {-1}^k \binom {r + k - 1} k\)

$\blacksquare$