## Weird binomial coefficients

The binomial coefficients find great many uses in combinatorics, but also in calculus. The usual way we understand the binomial coefficients is

$\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}$,

where $n$ and $k$ are integers. But what do you do with $\displaystyle \binom{\frac{1}{2}}{k}$?! Is it even defined?

While this is weird-looking, it is well-defined. If we write

$\displaystyle \binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}$,

which is strictly equivalent to the usual definition, but now depends more on $k$ than on $n$, it allows us to write…

$\displaystyle \binom{\frac{1}{2}}{n}=\frac{\frac{1}{2}( \frac{1}{2}-1)(\frac{1}{2}-2)\cdots(\frac{1}{2}-k+1)}{k!}$.

The first few $\displaystyle \binom{\frac{1}{2}}{k}$ are

$\displaystyle 1, \frac{1}{2}, -\frac{1}{8}, \frac{1}{16}, -\frac{5}{128}, \frac{7}{256}, -\frac{21}{1024}, \frac{33}{2048}, \ldots$.

### One Response to Weird binomial coefficients

1. […] expansion. The coefficients seems cumbersome: until we remember that we’ve already saw them before. This allows us to […]