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Ramanujan’s Most Beautiful Identity

G. H. Hardy called the following equation Ramanujan’s “most beautiful identity.” For |q| < 1,

\sum_{n=0}^\infty p(5n+4) q^n = 5 \prod_{n=1}^\infty \frac{(1 - q^{5n})^5}{(1 - q^n)^6}

If I understood it, I might say it’s beautiful, but for now I can only say it’s mysterious. Still, I explain what I can.

The function p on the left side is the partition function. For a positive integer argument n, p(n) is the number of ways one can write n as the sum of a non-decreasing sequence of positive integers.

The right side of the equation is an example of a q-series. Strictly speaking it’s a product, not a series, but it’s the kind of thing that goes under the general heading of q-series.

I hardly know anything about q-series, and they don’t seem very motivated. However, I keep running into them in unexpected places. They seem to be a common thread running through several things I’m vaguely familiar with and would like to understand better.

As mysterious as Ramanujan’s identity is, it’s not entirely unprecedented. In the eighteenth century, Euler proved that the generating function for partition numbers is a q-product:

\sum_{n=0}^\infty p(n) q^n = \prod_{n=1}^\infty \frac{1}{(1 - q^n)}

So in discovering his most beautiful identity (and others) Ramanujan followed in Euler’s footsteps.

Reference: An Invitation to q-series

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