Definition:Binomial Coefficient/Real Numbers

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Definition

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 $k \ge 0$:

$\displaystyle \binom r k = \frac {r \left({r - 1}\right) \cdots \left({r - k + 1}\right)} {k \left({k - 1}\right) \cdots 1} = \prod_{j \mathop = 1}^k \frac {r + 1 - j} j$

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.


Sources