Definition:Binomial Coefficient/Real Numbers

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

Then $\displaystyle \binom r k$ is defined as:
 * $\displaystyle \binom r k = \begin{cases}

\dfrac {r^{\underline k}} {k!} & : k \ge 0 \\ & \\ 0 & : k < 0 \end{cases}$ where $r^{\underline k}$ is defined as the falling factorial.

That is, when $k \ge 0$:
 * $\displaystyle \binom r k = \frac {r \left({r - 1}\right) \cdots \left({r - k + 1}\right)} {k \left({k - 1}\right) \cdots 1} = \prod_{j \mathop = 1}^k \frac {r + 1 - j} j$

It can be seen that this agrees with the definition for integers when $r$ is an integer.

For most applications the integer form is sufficient.