Rogers–Ramanujan continued fraction on arvau

Wikipedia / Rogers–Ramanujan continued fraction

The Rogers–Ramanujan continued fraction is a continued fraction discovered by 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

and,

with the coefficients of the q-expansion being and 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 (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. However, Ramanujan, in his examples to Hardy and given below, used the nomeinstead.

Special values

If q is the nome, are related to modular functions. Since they have integral coefficients, the theory of complex multiplication involving an imaginary quadratic field are algebraic numbers that can be evaluated explicitly.

Examples of R(q)

Given the general form where Ramanujan used the nome,

is the golden ratio is a positive root of the quartic equation,

are two positive roots of a single octic,

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

The algebraic degree k ().

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)

in terms of the j-function.

are algebraic numbers involving an imaginary quadratic field. For example, the formulas above simplify to,

and,

as the golden ratio.

Derivation of special values

Tangential sums

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: 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

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

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

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

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

, with even subscripts on the LHS. Thus,

Note, however, that theta functions normally use the nome , while the Dedekind eta function uses the square of the nome , thus the variable x. 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

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

Since also,

between the two equations, one can then express j(τ as,

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

where

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

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

Functional equation

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 the golden ratio,

Incidentally,

Modular equations

. Elegant ones for small prime n are as follows.

Other results

. as the golden ratio.

then,

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

where

, then,

Quintic equations

The general quintic equation in Bring-Jerrard form:

and the elliptic nome

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

Then the real solution is

Example 1

Transform to,

thus,

and the solution is:

and can not be represented by elementary root expressions.

Example 2