Rogers–Ramanujan identities

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.

Definition

The Rogers–Ramanujan identities are

G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = 
\frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}
	=1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \,
(sequence A003114 in OEIS)

and

H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = 
\frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
=1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots \,
(sequence A003106 in OEIS).

Here, (\cdot;\cdot)_n denotes the q-Pochhammer symbol.

Integer Partitions

Consider the following:

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 q1/60G(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+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\cdots}}}  = \frac{G(q)}{H(q)}.

See also

References

External links


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