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 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.
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 and q11/60H are modular functions of τ.
and Robert Lee Wilson were the first to prove Rogers–Ramanujan identities using completely representation-theoretictechniques. They proved these identities using level 3 modules for the affine Lie algebra. In the course of this proof they invented and used what they called -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 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.
