Rogers–Ramanujan identities
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and 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 . independently rediscovered and proved the identities. Definition. The Rogers–Ramanujan identities are and Here, formula_3 denotes the q-Pochhammer symbol. Combinatorial interpretation. Consider the following: The Rogers–Ramanujan identities could be now interpreted in the following way. Let formula_5  be a non-negative integer. Alternatively, Modular functions. If "q" = e, then "q""G"("q") and "q""H"("q") are modular functions of τ. Applications. The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. Ramanujan's continued fraction is Relations to affine Lie algebras and vertex operator algebras. James Lepowsky and Robert Lee Wilson were the first to prove Rogers–Ramanujan identities using completely representation-theoretic techniques. They proved these identities using level 3 modules for the affine Lie algebra formula_24. In the course of this proof they invented and used what they called formula_25 -algebras. Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels. It can be used to find (and prove) new partition identities. First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for the affine Lie algebra formula_26.