# Rogers–Ramanujan identities

In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series, first discovered and proved by Leonard James Rogers (1894). They were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities.

## Definition

The Rogers–Ramanujan identities are

$G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \,$ (sequence A003114 in OEIS)

and

$H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots \,$ (sequence A003106 in OEIS).

Here, $(\cdot;\cdot)_n$ denotes the q-Pochhammer symbol.

## Modular functions

If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ.

## A Framework for the Identities

In April 2014 Ken Ono, a number theorist at Emory University, announced that he had found a framework for the Rogers-Ramanujan identities and their arithmetic properties, solving a long-standing mystery stemming from the work of Ramanujan. The findings yield a treasure trove of algebraic numbers and formulas to access them. Ono has two co-authors for this work, S. Ole Warnaar of the University of Queensland and Michael Griffin, an Emory University graduate student.

## Applications

The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics.

$1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots}}} = \frac{G(q)}{H(q)}.$