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 […]

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