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.

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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%2B q %2Bq^2 %2Bq^3 %2B2q^4%2B2q^5 %2B3q^6%2B\cdots \,
(sequence A003114 in OEIS)

and

H(q) =\sum_{n=0}^\infty \frac {q^{n^2%2Bn}} {(q;q)_n} = 
\frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty}
=1%2Bq^2 %2Bq^3 %2Bq^4%2Bq^5 %2B2q^6%2B\cdots \,
(sequence A003106 in OEIS).

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

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%2B\frac{q}{1%2B\frac{q^2}{1%2B\frac{q^3}{1%2B\cdots}}}  = \frac{G(q)}{H(q)}.

See also

References

External links