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The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and independently by Srinivasa Ramanujan, and closely related to the Rogers–Ramanujan identities. It can be evaluated explicitly for a broad class of values of its argument.

Definition
Representation of the approximation $$q^{1/5}A_{400}(q)/B_{400}(q)$$ of the Rogers–Ramanujan continued fraction.

Given the functions G(q) and H(q) appearing in the Rogers–Ramanujan identities,

\begin{align}G(q) &= \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty}\\ &= \prod_{n=1}^\infty \frac{1}{(1-q^{5n-1})(1-q^{5n-4})}\\ &=\sqrt[60]{qj}\,_2F_1\left(-\tfrac{1}{60},\tfrac{19}{60};\tfrac{4}{5};\tfrac{1728}{j}\right)\\ &=\sqrt[60]{q\left(j-1728\right)}\,_2F_1\left(-\tfrac{1}{60},\tfrac{29}{60};\tfrac{4}{5};-\tfrac{1728}{j-1728}\right)\\ &= 1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots \end{align}

and,

\begin{align}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}\\ &= \prod_{n=1}^\infty \frac{1}{(1-q^{5n-2})(1-q^{5n-3})}\\ &=\frac{1}{\sqrt[60]{q^{11}j^{11}}}\,_2F_1\left(\tfrac{11}{60},\tfrac{31}{60};\tfrac{6}{5};\tfrac{1728}{j}\right)\\ &=\frac{1}{\sqrt[60]{q^{11}\left(j-1728\right)^{11}}}\,_2F_1\left(\tfrac{11}{60},\tfrac{41}{60};\tfrac{6}{5};-\tfrac{1728}{j-1728}\right)\\ &= 1+q^2 +q^3 +q^4+q^5 +2q^6+2q^7+\cdots \end{align}

OEIS A003114 and OEIS A003106, respectively, where (a;q)_\infty denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function, then the Rogers–Ramanujan continued fraction is,

\begin{align}R(q) &= \frac{q^{\frac{11}{60}}H(q)}{q^{-\frac{1}{60}}G(q)} = q^{\frac{1}{5}}\prod_{n=1}^\infty \frac{(1-q^{5n-1})(1-q^{5n-4})}{(1-q^{5n-2})(1-q^{5n-3})}\\ &= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\ddots}}}} \end{align}

Modular functions

If $$q=e^{2\pi{\rm{i}}\tau}$$ , then $$q^{-\frac{1}{60}}G(q)$$ and $$q^{\frac{11}{60}}H(q),$$ as well as their quotient R(q), are modular functions of $$\tau$$ . Since they have integral coefficients, the theory of complex multiplication implies that their values for $$\tau$$ an imaginary quadratic irrational are algebraic numbers that can be evaluated explicitly.

Examples

$$R\big(e^{-2\pi}\big) = \cfrac{e^{-\frac{2\pi}{5}}}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\ddots}}} = {\sqrt{5+\sqrt{5}\over 2}-{1+\sqrt{5}\over 2}}$$

$$R\big(e^{-2\sqrt{5}\pi}\big) = \cfrac{e^{-\frac{2\pi}{\sqrt5}}}{1+\cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1+\ddots}}} = \frac{\sqrt{5}}{1+\big(5^{3/4} (\phi-1)^{5/2}-1\big)^{1/5}} - {\phi}$$

where$$\phi=\frac{1+\sqrt5}{2}$$ is the golden ratio.

Relation to modular forms

It can be related to the Dedekind eta function, a modular form of weight 1/2, as,[1]

$$\frac{1}{R(q)}-R(q) = \frac{\eta(\frac{\tau}{5})}{\eta(5\tau)}+1$$

$$\frac{1}{R^5(q)}-R^5(q) = \left[\frac{\eta(\tau)}{\eta(5\tau)}\right]^6+11$$

Relation to j-function

Among the many formulas of the j-function, one is,

$$j(\tau) = \frac{(x^2+10x+5)^3}{x}$$

where,

$$x = \left[\frac{\sqrt{5}\,\eta(5\tau)}{\eta(\tau)}\right]^6$$

Eliminating the eta quotient, one can then express j(τ) in terms of r =R(q) as,

$$j(\tau) = -\frac{(r^{20}-228r^{15}+494r^{10}+228r^5+1)^3}{r^5(r^{10}+11r^5-1)^5}$$

$$j(\tau)-1728 = -\frac{(r^{30}+ 522r^{25}- 10005 r^{20}- 10005 r^{10}- 522 r^{5}+1)^2}{r^5(r^{10}+ 11 r^5-1)^5}$$

where the numerator and denominator are polynomial invariants of the icosahedron. Using the modular equation between R(q) and $$R(q^5)$$ , one finds that,

$$j(5\tau) = -\frac{(r^{20}+12r^{15}+14r^{10}-12r^5+1)^3}{r^{25}(r^{10}+11r^5-1)}$$

let $$z=r^5-\frac{1}{r^5}$$ ,then $$j(5\tau) = -\frac{\left(z^2+12z+16\right)^3}{z+11}$$

where,

$$z_{\infty}= -\left[\frac{\sqrt{5}\,\eta(25\tau)}{\eta(5\tau)}\right]^6-11,z_0=-\left[\frac{\eta(\tau)}{\eta(5\tau)}\right]^6-11,z_1=\left[\frac{\eta(\frac{5\tau+2}{5})}{\eta(5\tau)}\right]^6-11,z_2=-\left[\frac{\eta(\frac{5\tau+4}{5})}{\eta(5\tau)}\right]^6-11,z_3=\left[\frac{\eta(\frac{5\tau+6}{5})}{\eta(5\tau)}\right]^6-11,z_4=-\left[\frac{\eta(\frac{5\tau+8}{5})}{\eta(5\tau)}\right]^6-11$$

which in fact is the j-invariant of the elliptic curve,

$$y^2+(1+r^5)xy+r^5y=x^3+r^5x^2$$

parameterized by the non-cusp points of the modular curve $$X_1(5).$$

Functional equation

For convenience, one can also use the notation $$r(\tau) = R(q)$$ when q = e2πiτ. While other modular functions like the j-invariant satisfies,

$$j(-\tfrac{1}{\tau}) = j(\tau)$$

and the Dedekind eta function has,

$$\eta(-\tfrac{1}{\tau}) =\sqrt{-i\tau}\, \eta(\tau)$$

the functional equation of the Rogers–Ramanujan continued fraction involves[2] the golden ratio \phi,

$$r(-\tfrac{1}{\tau}) = \frac{1-\phi\,r(\tau)}{\phi+r(\tau)}$$

Incidentally,

$$r(\tfrac{7+i}{10}) = i$$

Modular equations

There are modular equations between R(q) and $$R(q^n)$$. Elegant ones for small prime n are as follows.[3]

For n = 2, let u=R(q) and $$v=R(q^2)$$ , then $$v-u^2 = (v+u^2)uv^2.$$

For n = 3, let u=R(q) and$$v=R(q^3)$$ , then $$(v-u^3)(1+uv^3) = 3u^2v^2.$$

For n = 5, let u=R(q) and $$v=R(q^5), then\( (v^4-3v^3+4v^2-2v+1)v=(v^4+2v^3+4v^2+3v+1)u^5. For n = 11, let u=R(q) and \( v=R(q^{11})$$ , then $$uv(u^{10}+11u^5-1)(v^{10}+11v^5-1) = (u-v)^{12}.$$

Regarding n = 5, note that $$v^{10}+11v^5-1=(v^2+v-1)(v^4-3v^3+4v^2-2v+1)(v^4+2v^3+4v^2+3v+1).$$

Other results

Ramanujan found many other interesting results regarding R(q).[4] Let $$u=R(q^a), v=R(q^b)$$ , and $$\phi$$ as the golden ratio.

If $$ab=4\pi^2$$ , then $$(u+\phi)(v+\phi)=\sqrt{5}\,\phi.$$

If $$5ab=4\pi^2$$ , then $$(u^5+\phi^5)(v^5+\phi^5)=5\sqrt{5}\,\phi^5.$$

The powers of R(q) also can be expressed in unusual ways. For its cube,

$$R^3(q) = \frac{\sum_{n=0}^\infty\frac{q^{2n}}{1-q^{5n+2}}-\sum_{n=0}^\infty\frac{q^{3n+1}}{1-q^{5n+3}} }{\sum_{n=0}^\infty\frac{q^{n}}{1-q^{5n+1}}-\sum_{n=0}^\infty\frac{q^{4n+3}}{1-q^{5n+4}} }$$

For its fifth power, let $$w=R(q)R^2(q^2)$$ , then,

$$R^5(q) = w\left(\frac{1-w}{1+w}\right)^2,\;\; R^5(q^2) = w^2\left(\frac{1+w}{1-w}\right)$$

References

Duke, W. "Continued Fractions and Modular Functions", http://www.math.ucla.edu/~wdduke/preprints/bams4.pdf
Duke, W. "Continued Fractions and Modular Functions" (p.9)
Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction", http://www.math.uiuc.edu/~berndt/articles/rrcf.pdf

Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"

Rogers, L. J. (1894), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., s1-25 (1): 318–343, doi:10.1112/plms/s1-25.1.318
Berndt, B. C.; Chan, H. H.; Huang, S. S.; Kang, S. Y.; Sohn, J.; Son, S. H. (1999), "The Rogers–Ramanujan continued fraction" (PDF), Journal of Computational and Applied Mathematics 105: 9, doi:10.1016/S0377-0427(99)00033-3