Rogers–Ramanujan identities

Remarks

In terms of integer partitions, the identities say the following for each $m \ge 0$. The partitions of $m$ whose parts differ pairwise by at least $2$ are as many as the partitions of $m$ into parts congruent to $1$ or $4$ modulo $5$. Consider the partitions of $m$ whose parts differ by at least $2$ and are all at least $2$. They are as many as the partitions of $m$ into parts congruent to $2$ or $3$ modulo $5$. The term $q^{n^2} / (q; q)_n$ counts the partitions of the first kind with exactly $n$ parts, since $n^2 = 1 + 3 + \cdots + (2n - 1)$ is the smallest such sum.

Example: for $m = 6$, the partitions with parts differing by at least $2$ are $6$, $5 + 1$ and $4 + 2$. The partitions into parts from $\set{1, 4, 6, 9, \ldots}$ are $6$, $4 + 1 + 1$ and $1 + 1 + 1 + 1 + 1 + 1$. Both counts are $3$.

A standard proof transforms the sum side and then applies the Jacobi triple product to reach the product side. The identities were found by Leonard James Rogers and later found again by Srinivasa Ramanujan.

Going deeper

For a student

The identities are the prototype of partition identities of Rogers–Ramanujan type, where a difference condition on the parts matches a congruence condition. Their sum sides are basic hypergeometric series and their product sides are, up to a power of $q$, quotients of theta functions. They appear in the theory of modular forms, in the hard hexagon model of statistical mechanics, and in the representation theory of affine Lie algebras. Compare with Euler's partition theorem, where parts are distinct on one side and odd on the other.