I was skimming through the book Mathematical Reflections [1] recently. He was discussing a set of generalizations [2] of the Star of David theorem from combinatorics.

gcdleft(binom{n - 1}{r - 1}, binom{n}{r+1}, binom{n+1}{r}right) = gcdleft(binom{n-1}{r}, binom{n+1}{r+1}, binom{n}{r-1}right)

The theorem is so named because if you draw a Star of David by connecting points in Pascal’s triangle then each side corresponds to the vertices of a triangle.

diagram illustrating the Star of David Theorem

One such theorem was the following.

binom{n - ell}{r - ell} binom{n - k}{r} binom{n - k - ell}{r - k} = binom{n-k}{r-k} binom{n - ell}{r} binom{n - k - ell}{r - ell}

This theorem also has a geometric interpretation, connecting vertices within Pascal’s triangle.

The authors point out that the binomial coefficient is a separable function of three variables, and that their generalized Star of David theorem is true for any separable function of three variables.

The binomial coefficient C(n, k) is a function of two variables, but you can think of it as a function of three variables: n, k, and n − k. That is

{n choose k} = f(n) , g(k) , g(n-k)

where f(n) = n! and g(k) = 1/k!.

I was surprised to see the term separable function outside of a PDE context. My graduate work was in partial differential equations, and so when I hear separable function my mind goes to separation of variables as a technique for solving PDEs.

Coincidentally, I was looking a separable coordinate systems recently. These are coordinate systems in which the Helmholtz equation can be solved by separable function, i.e. a coordinate system in which the separation of variables technique will work. The Laplacian can take on very different forms in different coordinate systems, and if possible you’d like to choose a coordinate system in which a PDE you care about is separable.

Related posts

[1] Peter Hilton, Derek Holton, and Jean Pedersen. Mathematical Reflections. Springer, 1996.

[2] Hilton et al refer to a set of theorems as generalizations of the Star of David theorem, but these theorems are not strictly generalizations in the sense that the original theorem is clearly a special case of the generalized theorems. The theorems are related, and I imagine with more effort I could see how to prove the older theorem from the newer ones, but it’s not immediately obvious.

The post Separable functions in different contexts first appeared on John D. Cook.

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