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.

Domain coloring representation of the convergent

A_{{400}}(q)/B_{{400}}(q)

of the function

q^{{-1/5}}R(q)

, where

R(q)

is the Rogers–Ramanujan continued fraction.

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{aligned}G(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\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}}\,_{2}F_{1}\left(-{\tfrac {1}{60}},{\tfrac {19}{60}};{\tfrac {4}{5}};{\tfrac {1728}{j}}\right)\\&={\sqrt[{60}]{q\left(j-1728\right)}}\,_{2}F_{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{aligned}}

and,

{\begin{aligned}H(q)&=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(1-q)(1-q^{2})\cdots (1-q^{n})}}=\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}}}}\,_{2}F_{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}}}}\,_{2}F_{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{aligned}}

A003114 and 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{aligned}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{aligned}}

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}-\phi}

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

{\begin{aligned}&j(\tau )=-{\frac {(r^{20}-228r^{15}+494r^{10}+228r^{5}+1)^{3}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\\[6pt]&j(\tau )-1728=-{\frac {(r^{30}+522r^{25}-10005r^{20}-10005r^{10}-522r^{5}+1)^{2}}{r^{5}(r^{10}+11r^{5}-1)^{5}}}\end{aligned}}

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

{\begin{aligned}&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,\\[6pt]&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\end{aligned}}

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

y^{2}+(1+r^{5})xy+r^{5}y=x^{3}+r^{5}x^{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 = e^2πiτ. 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 $(v-u^{3})(1+uv^{3})=3u^{2}v^{2}.$

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 {\alpha }{\beta }}

where,

\alpha =\sum _{n=0}^{\infty }{\frac {q^{2n}}{1-q^{5n+2}}}-\sum _{n=0}^{\infty }{\frac {q^{3n+1}}{1-q^{5n+3}}}

\beta =\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

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[1] Duke, W. "Continued Fractions and Modular Functions", http://www.math.ucla.edu/~wdduke/preprints/bams4.pdf

[2] Duke, W. "Continued Fractions and Modular Functions" (p.9)

[3] Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction", http://www.math.uiuc.edu/~berndt/articles/rrcf.pdf

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