Rogers–Ramanujan continued fraction

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

External links

Weisstein, Eric W., "Rogers-Ramanujan Identities", MathWorld.
Weisstein, Eric W., "Rogers-Ramanujan Continued Fraction", MathWorld.

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