\begin{definition}[Definition:Binomial Coefficient/Real Numbers]
Let $r \in \R, k \in \Z$.
Then $\dbinom r k$ is defined as:
:$\dbinom r k = \begin {cases}
\dfrac {r^{\underline k} } {k!} & : k \ge 0 \\
& \\
0 & : k < 0
\end {cases}$
where $r^{\underline k}$ denotes the falling factorial.
That is, when $k \ge 0$:
:$\ds \dbinom r k = \dfrac {r \paren {r - 1} \cdots \paren {r - k + 1} } {k \paren {k - 1} \cdots 1} = \prod_{j \mathop = 1}^k \dfrac {r + 1 - j} j$
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.
\end{definition}