Falling power analog of binomial theorem

Yesterday I wrote about how the right notation could make Newton's interpolation theorem much easier to remember, revealing it as an analog of Taylor series. This post will do something similar for the binomial theorem.

Let's start with the following identity.

It's not clear that this is true, or how one might generalize it. But if we rewrite the equation using falling power notation we have

which looks a lot like the binomial theorem. In fact it is the case n = 3 of the Chu-Vandermonde theorem which says

Viewed purely visually, this is the binomial theorem with little lines under each exponent.

Incidentally, the analogous theorem holds for rising powers. Just change all the lines under the exponents to lines on top of the exponents.

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