Ramanujan's AGM continued fraction

The Rogers-Ramanujan continued fraction is a generalized continued fraction defined by R(q) congruent q^(1/5)/(1 + q/(1 + q^2/(1 + q^3/(1 + ...)))) (Rogers 1894, Ramanujan 1957, Berndt et al. 1996, 1999, 2000). It was discovered by Rogers, independently by Ramanujan around 1913, and again independently by Schur in 1917. Modulo the factor of q^(1/5) added for convenience, it provides a geometric series q-analog of the golden ratio ϕ = 1 + 1/(1 + 1/(1 + 1/(1 + ...))).

Bauer-Muir transformation | fountain | generalized continued fraction | golden ratio | q-series | Ramanujan theta functions | Rogers-Ramanujan identities