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.

denotes the infinite q-Pochhammer symbol, j is the j-function, and 2F1 is the hypergeometric function.

The Rogers–Ramanujan continued fraction is then 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) uses

However, Ramanujan, in his examples to Hardy and given below, used the nome

[citation needed] If q is the nome or its square, then

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.

Given the general form where Ramanujan used the nome

is a positive root of the quartic equation, while

are two positive roots of a single octic, (since

has a square root) which explains the similarity of the two closed-forms.

are two roots of the same equation as well as, The algebraic degree k of

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

are algebraic numbers (though normally of high degree) for

involving an imaginary quadratic field.

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: 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: 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:

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.

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

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.

Ramanujan found many other interesting results regarding

, then, The general quintic equation in Bring-Jerrard form: for every real value

can be solved in terms of Rogers-Ramanujan continued fraction

Recall in the previous section the 5th power of

: Transform to, thus, and the solution is: and can not be represented by elementary root expressions.

thus, Given the more familiar continued fractions with closed-forms, with golden ratio