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 of the function , where is the Rogers–Ramanujan continued fraction.

Definition

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

Given the functions   and   appearing in the Rogers–Ramanujan identities, and assume  ,

 

and,

 

with the coefficients of the q-expansion being OEISA003114 and OEISA003106, respectively, where   denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function. The Rogers–Ramanujan continued fraction is then

 
  is the Jacobi symbol.

One should be careful with notation since the formulas employing the j-function   will be consistent with the other formulas only if   (the square of the nome) is used throughout this section since the q-expansion of the j-function (as well as the well-known Dedekind eta function) uses  . However, Ramanujan, in his examples to Hardy and given below, used the nome  instead.[citation needed]

Special values

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If q is the nome or its square, then   and  , as well as their quotient  , are related to modular functions of  . Since they have integral coefficients, the theory of complex multiplication implies that their values for   involving an imaginary quadratic field are algebraic numbers that can be evaluated explicitly.

Examples of R(q)

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Given the general form where Ramanujan used the nome  ,

 

f when  ,

 

when  ,

 

when  ,

 

when  ,

 

when  ,

 

when  ,

 

when  ,

 

and   is the golden ratio. Note that   is a positive root of the quartic equation,

 

while   and   are two positive roots of a single octic,

 

(since   has a square root) which explains the similarity of the two closed-forms. More generally, for positive integer m, then   and   are two roots of the same equation as well as,

 

The algebraic degree k of   for   is   (OEISA082682).

Incidentally, these continued fractions can be used to solve some quintic equations as shown in a later section.

Examples of G(q) and H(q)

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Interestingly, there are explicit formulas for   and   in terms of the j-function   and the Rogers-Ramanujan continued fraction  . However, since   uses the nome's square  , then one should be careful with notation such that   and   use the same  .

 
 

Of course, the secondary formulas imply that   and   are algebraic numbers (though normally of high degree) for   involving an imaginary quadratic field. For example, the formulas above simplify to,

 

and,

 

and so on, with   as the golden ratio.

Derivation of special values

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Tangential sums

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In the following we express the essential theorems of the Rogers-Ramanujan continued fractions R and S by using the tangential sums and tangential differences:

 
 

The elliptic nome and the complementary nome have this relationship to each other:

 

The complementary nome of a modulus k is equal to the nome of the Pythagorean complementary modulus:

 

These are the reflection theorems for the continued fractions R and S:

 
 

The letter   represents the Golden number exactly:

 
 

The theorems for the squared nome are constructed as follows:

 
 

Following relations between the continued fractions and the Jacobi theta functions are given:

 
 

Derivation of Lemniscatic values

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Into the now shown theorems certain values are inserted:

 

Therefore following identity is valid:

 

In an analogue pattern we get this result:

 

Therefore following identity is valid:

 

Furthermore we get the same relation by using the above mentioned theorem about the Jacobi theta functions:

 
 

This result appears because of the Poisson summation formula and this equation can be solved in this way:

 

By taking the other mentioned theorem about the Jacobi theta functions a next value can be determined:

 
 

That equation chain leads to this tangential sum:

 

And therefore following result appears:

 

In the next step we use the reflection theorem for the continued fraction R again:

 
 

And a further result appears:

 

Derivation of Non-Lemniscatic values

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The reflection theorem is now used for following values:

 

The Jacobi theta theorem leads to a further relation:

 
 

By tangential adding the now mentioned two theorems we get this result:

 
 

By tangential substraction that result appears:

 
 

In an alternative solution way we use the theorem for the squared nome:

 
 

Now the reflection theorem is taken again:

 
 

The insertion of the last mentioned expression into the squared nome theorem gives that equation:

 

Erasing the denominators gives an equation of sixth degree:

 
 

The solution of this equation is the already mentioned solution:

 

Relation to modular forms

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

 
 

The Rogers-Ramanujan continued fraction can also be expressed in terms of the Jacobi theta functions. Recall the notation,

 

The notation   is slightly easier to remember since  , with even subscripts on the LHS. Thus,

 
 
 
 

Note, however, that theta functions normally use the nome q = eiπτ, while the Dedekind eta function uses the square of the nome q = e2iπτ, thus the variable x has been employed instead to maintain consistency between all functions. For example, let   so  . Plugging this into the theta functions, one gets the same value for all three R(x) formulas which is the correct evaluation of the continued fraction given previously,

 

One can also define the elliptic nome,

 

The small letter k describes the elliptic modulus and the big letter K describes the complete elliptic integral of the first kind. The continued fraction can then be also expressed by the Jacobi elliptic functions as follows:

 

with

 

Relation to j-function

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One formula involving the j-function and the Dedekind eta function is this:

 

where   Since also,

 

Eliminating the eta quotient   between the two equations, 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

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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

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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  


Or equivalently for  , let   and   and  , then  


For  , let   and  , then  


Regarding  , note that  

Other results

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

If   then,

 

If   then,

 

The powers of   also can be expressed in unusual ways. For its cube,

 

where

 
 

For its fifth power, let  , then,

 

Quintic equations

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The general quintic equation in Bring-Jerrard form:

 

for every real value   can be solved in terms of Rogers-Ramanujan continued fraction   and the elliptic nome

 

To solve this quintic, the elliptic modulus must first be determined as

 

Then the real solution is

 

where  . Recall in the previous section the 5th power of   can be expressed by  :

 

Example 1

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Transform to,

 

thus,

 
 
 
 
 

and the solution is:

 

and can not be represented by elementary root expressions.

Example 2

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thus,

 

Given the more familiar continued fractions with closed-forms,

 
 
 

with golden ratio   and the solution simplifies to

 

References

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  1. ^ Duke, W. "Continued Fractions and Modular Functions", https://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"
  • 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 (1–2): 9–24, doi:10.1016/S0377-0427(99)00033-3
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