In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series

Basic hypergeometric series

In mathematics, Heine's basic hypergeometric series, or hypergeometric q-series, are q-analog generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series....

, first discovered and proved by . They were subsequently rediscovered (without a proof) by Srinivasa Ramanujan

Srinivasa Ramanujan

Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof . independently rediscovered and proved the identities.

Definition

The Rogers–Ramanujan identities are

and
.

Here, denotes the q-Pochhammer symbol.

Modular functions

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

Applications

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

Hard hexagon model

In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent....

 in statistical mechanics.

Ramanujan's continued fraction is