Rogers–Ramanujan continued fraction

The Rogers–Ramanujan continued fraction is a continued fraction discovered by Rogers (1894) and later studied by Srinivasa Ramanujan, closely related to the Rogers-Ramanujan identities, that can be evaluated explicitly for special values of its argument.

Definition

Ramanujan's continued fraction is

$1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}} = \frac{G(q)}{H(q)}=1+q -q^3 +q^5-\cdots$ (sequence A003823 in OEIS)

where

$G(q) = \sum_{n=0}^\infty \frac {q^{n^2}} {(q;q)_n} = \frac {1}{(q;q^5)_\infty (q^4; q^5)_\infty} =1+ q +q^2 +q^3 +2q^4+2q^5 +3q^6+\cdots$ (sequence A003114 in OEIS)

and

$H(q) =\sum_{n=0}^\infty \frac {q^{n^2+n}} {(q;q)_n} = \frac {1}{(q^2;q^5)_\infty (q^3; q^5)_\infty} =1+q^2 +q^3 +q^4+q^5 +2q^6+\cdots.$ (sequence A003106 in OEIS)

are the functions appearing in the Rogers-Ramanujan identities.

Here, $(a;q)_\infty$ denotes the infinite q-Pochhammer symbol.

Modular functions

If q = e^2πiτ, then q^−1/60G(q) and q^11/60H(q) and therefore q^1/5H(q)/G(q)) are modular functions of τ. Since they have integral coefficients, the theory of complex multiplication implies that their values for τ an imaginary quadratic irrational are algebraic numbers that can be evaluated explicitly. In particular Ramanujan's continued fraction can be evaluated for these values of τ.

Examples

$\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\varphi\sqrt{5}}-\varphi}\right) = 0.9981360\dots$

where φ is the golden ratio (Approximately 1.618)

The multiplicative inverse of this expression is:

$\begin{align} & {} \quad 1 + \cfrac{e^{-2\pi}}{1+ \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1+\dots}}} = \frac{1}{2}\left[1+\sqrt{5}+\sqrt{2(5+\sqrt{5})}\right]\,e^{-2\pi/5} \\ \\ & = \frac{e^{-2\pi/5}}{\sqrt{\varphi\sqrt{5}} - \varphi } = 1.0018674\dots \end{align}$

$\begin{align} & {} \quad \cfrac{1}{1+\cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1+\dots}}} \\ \\ & = \left( \frac{\sqrt{5}}{1+[5^{3/4} (\varphi-1)^{5/2}-1]^{1/5}} - {\varphi}\right) \, e^{2\pi/\sqrt{5}} = 0.99999920\dots \end{align}$

The multiplicative inverse of this expression is:

$\begin{align} & {} \quad 1 + \cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1+\dots}} \\ \\ & = \cfrac{e^{-2\pi/\sqrt{5}}}{{}\ \ \cfrac{\sqrt{5}}{ 1+\left[5^{3/4}( \varphi-1)^{5/2} - 1\right]^{1/5}} - \varphi\ \ {}} = 1.000000791267\dots \end{align}$

References

• 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 
• Bruce C. Berndt, Heng Huat Chan,, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24.

External links

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