One shouldn’t disturb binomial identities too much

When teaching the most elementary courses at the university, a little bit of combinatorics enters, and the relation between the binomial coefficients and the expansion of (x+y)k, for non-negative integers k. An often appreciated trick for the students is to prove some identities among binomial coefficients using “analytic” properties of polynomials, and interpret them combinatorially, or conversely. The most basic of these identities is probably

id-f1.png

In the spirit of fun, assume we think of rewriting this as

id-f2.png

And now, maybe while whistling idly to pretend that we are not doing anything, let’s jolt the denominators with a quick flick of the finger:

id-f3.png

Since it seems that no one has noticed anything, let’s do it again:

id-f4.png

and again, and again… but stop! a red-faced policeman comes, and says that he has nothing against some good clean fun, especially on Boat Race Night, but enough is enough, and what horror have we done with poor Newt’s lovely identity:

id-f5.png

Well, of course, what is behind this is an undoubtedly well-known identity, which can be expressed in terms of hypergeometric functions with very general parameters. But the sliding denominators might be a nice thing to show students, and the (or at least, one elementary one) actual proof of the actual formula is a good exercise in exploiting “polynomiality” in various forms, so it can be used for that purpose…

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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