Over a million developers have joined DZone.
Platinum Partner

Ramanujan’s Most Beautiful Identity

· Big Data Zone

The Big Data Zone is presented by Exaptive.  Learn how rapid data application development can address the data science shortage.

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

The Big Data Zone is presented by Exaptive.  Learn about how to rapidly iterate data applications, while reusing existing code and leveraging open source technologies.

Topics:

Published at DZone with permission of John Cook , DZone MVB .

Opinions expressed by DZone contributors are their own.

{{ parent.title || parent.header.title}}

{{ parent.tldr }}

{{ parent.urlSource.name }}