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, denotes the q-Pochhammer symbol. Combinatorial interpretation Consider the following: is the generating function for partitions with exactly parts such that adjacent parts have difference at least 2. is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5. is the generating function for partitions with exactly parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2. is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5. The Rogers–Ramanujan identities could be now interpreted in the following way. Let be a non-negative integer. The number of partitions of such that the adjacent parts differ by at least 2 is the same as the number of partitions of such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of such that each part is congruent to either 2 or 3 modulo 5. Alternatively, The number of partitions of such that with parts the smallest part is at least is the same as the number of partitions of such that each part is congruent to either 1 or 4 modulo 5. The number of partitions of such that with parts the smallest part is at least is the same as the number of partitions of such that each part is congruent to either 2 or 3 modulo 5. Modular functions If q = e2πiτ, then q−1/60G(q) and q11/60H(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