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.

The Rogers–Ramanujan identities are

and

Here, ${\displaystyle (\cdot ;\cdot )_{n}}$ denotes the q-Pochhammer symbol.

Consider the following:

The Rogers–Ramanujan identities could be now interpreted in the following way. Let ${\displaystyle n}$ be a non-negative integer.

Alternatively,

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) are modular functions of τ.

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