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Rogers–Ramanujan identities

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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.

Contents

Definition

The Rogers–Ramanujan identities are

G ( q ) = ∑ n = 0 ∞ q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + ⋯ (sequence A003114 in the OEIS)

and

H ( q ) = ∑ n = 0 ∞ q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + ⋯ (sequence A003106 in the OEIS).

Here, ( ⋅ ; ⋅ ) n denotes the q-Pochhammer symbol.

Integer Partitions

Consider the following:

  • q n 2 ( q ; q ) n is the generating function for partitions with exactly n parts such that adjacent parts have difference at least 2.
  • 1 ( q ; q 5 ) ∞ ( q 4 ; q 5 ) ∞ is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5.
  • q n 2 + n ( q ; q ) n is the generating function for partitions with exactly n parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.
  • 1 ( q 2 ; q 5 ) ∞ ( q 3 ; q 5 ) ∞ 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 n be a non-negative integer.

    1. The number of partitions of n such that the adjacent parts differ by at least 2 is the same as the number of partitions of n such that each part is congruent to either 1 or 4 modulo 5.
    2. The number of partitions of n 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 n such that each part is congruent to either 2 or 3 modulo 5.

    Alternatively,

    1. The number of partitions of n such that with k parts the smallest part is at least k is the same as the number of partitions of n such that each part is congruent to either 1 or 4 modulo 5.
    2. The number of partitions of n such that with k parts the smallest part is at least k + 1 is the same as the number of partitions of n 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

    1 + q 1 + q 2 1 + q 3 1 + ⋯ = G ( q ) H ( q ) .

    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 s l 2 ^ . In the course of this proof they invented and used what they called Z -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 A 2 ( 2 ) .

    References

    Rogers–Ramanujan identities Wikipedia
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