Rogers–Ramanujan continued fraction

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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.

Domain coloring representation of the convergent of the function , where is the Rogers–Ramanujan continued fraction.

Definition[edit]

Representation of the approximation of the Rogers–Ramanujan continued fraction.

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

and,

OEISA003114 and OEISA003106, respectively, where denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function, then the Rogers–Ramanujan continued fraction is,

Modular functions[edit]

If , then and , as well as their quotient , are modular functions of . Since they have integral coefficients, the theory of complex multiplication implies that their values for an imaginary quadratic irrational are algebraic numbers that can be evaluated explicitly.

Examples[edit]


where is the golden ratio.

Relation to modular forms[edit]

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

Relation to j-function[edit]

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

where

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

where the numerator and denominator are polynomial invariants of the icosahedron. Using the modular equation between and , one finds that,

let ,then

where

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

parameterized by the non-cusp points of the modular curve .

Functional equation[edit]

For convenience, one can also use the notation when q = e2πiτ. While other modular functions like the j-invariant satisfies,

and the Dedekind eta function has,

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

Incidentally,

Modular equations[edit]

There are modular equations between and . Elegant ones for small prime n are as follows.[3]

For , let and , then


For , let and , then


For , let and , then


For , let and , then


Regarding , note that

Other results[edit]

Ramanujan found many other interesting results regarding R(q).[4] Let , , and as the golden ratio.

If , then
If , then

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

where,

For its fifth power, let , then,

References[edit]

  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"

External links[edit]