Negated Upper Index of Binomial Coefficient

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Theorem

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

Then:

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

where $\dbinom r k$ is a binomial coefficient.


Corollary 1

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.


Corollary 2

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


Complex Numbers

For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $t, w$ integers:

$\dbinom z w = \dfrac {\map \sin {\pi \paren {w - z - 1} } } {\map \sin {\pi z} } \dbinom {w - z - 1} w$

where $\dbinom z w$ is a binomial coefficient.


Proof

\(\ds \binom r k\) \(=\) \(\ds \frac {r^{\underline k} } {k!}\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds \frac 1 {k!} \prod_{j \mathop = 0}^{k - 1} \paren {r - j}\) Definition of Falling Factorial
\(\ds \) \(=\) \(\ds \frac {\paren {-1}^k} {k!} \prod_{j \mathop = 0}^{k - 1} \paren {-\paren {r - j} }\)
\(\ds \) \(=\) \(\ds \frac {\paren {-1}^k} {k!} \prod_{j \mathop = 0}^{k - 1} \paren {j - r}\)
\(\ds \) \(=\) \(\ds \frac {\paren {-1}^k} {k!} \prod_{j \mathop = 0}^{k - 1} \paren {\paren {k - 1} - j - r}\) Permutation of Indices of Product
\(\ds \) \(=\) \(\ds \frac {\paren {-1}^k} {k!} \prod_{j \mathop = 0}^{k - 1} \paren {\paren {k - r - 1} - j}\)
\(\ds \) \(=\) \(\ds \paren {-1}^k \frac {\paren {k - r - 1}^{\underline k} } {k!}\) Definition of Falling Factorial
\(\ds \) \(=\) \(\ds \paren {-1}^k \binom {k - r - 1} k\) Definition of Binomial Coefficient

$\blacksquare$


Sources

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