In his most recent post, Prof. Tao shared the following elementary discovery:

He first discovered this empirically in the course of his research, and then found a short proof. He didn’t share the proof on his blog yet, and instead challenged the readers to find their own proofs and observations. In this short post I’ll share a nice solution of mine. (There are some partial solutions I might add later, if I’ll be able to complete them).

The key is Leibniz’ rule. We begin by writing as a product: . We need to differentiate this product times, and we do this by applying Leibniz’ rule repeatedly:

(The notation stands for “falling factorial”: .)

The crucial observation is that:

Given this observation (which we’ll explain shortly), the even case is immediate and the odd case follows directly from the binomial theorem:

To explain the observation, note that in the even case contains the term whenever . In the odd case we need to write explicitly the terms in all 3 expressions (which are given as products), and seeing that they are the same (up to reordering and\or sign) – if we assume (w.l.o.g) we find:

And that:

There’s no clever combinatorial explanation\algebraic manipulation needed. This is why I think of this as a tautological identity, although it is surprising at first sight.

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## About Ofir Gorodetsky

Graduate student at TAU.
Can be contacted at bambaman1 at gmail dot com.

The above observation can be verified also by writing each of the three “failing factorials” in this observation as a ratio between two factorials (with fractional arguments) and than using the identity whenever .