---
tag: 7A68BZR6QS
kind: concept
name: Rogers–Ramanujan continued fraction
msc: [11A55, 11P84, "11F03"]
specializes: [04P2RJ4WPS]
sources:
  - url: https://en.wikipedia.org/wiki/Rogers–Ramanujan_continued_fraction
same_as:
  wikidata: Q7359378
  wikipedia: Rogers–Ramanujan continued fraction
---

## Summary

The Rogers–Ramanujan continued fraction is $R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^2}{1 + \cdots}}}$ for $\abs{q} < 1$.
It equals $q^{1/5}$ times the quotient of the two series of the [[Rogers–Ramanujan identities|M24H4GJPPW]], hence an infinite product.
As a function of $\tau$ with $q = e^{2\pi i \tau}$ it is a [[modular function|63TX60RTRQ]] for the group $\Gamma(5)$.
Ramanujan found closed forms of its values at special points, such as $R(e^{-2\pi}) = \sqrt{\frac{5 + \sqrt5}{2}} - \frac{1 + \sqrt5}{2}$.

## Explanations

### L1 human

A [[continued fraction|04P2RJ4WPS]] is a fraction whose denominator contains a fraction, whose denominator contains a fraction, and so on.
In this one the numerators are the powers $q, q^2, q^3, \dots$ and the extra factor $q^{1/5}$ in front makes the function behave well under symmetries.
At $q = e^{-2\pi}$ the value is an explicit number built from $\sqrt5$, related to the [[golden ratio|0SC8T7548D]].
Ramanujan sent such evaluations to Hardy in his famous letters.