Rogers–Ramanujan identities

The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913.

Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919).

Issai Schur (1917) independently rediscovered and proved the identities.

Alternatively, Since the terms occurring in the identity are generating functions of certain partitions, the identities make statements about partitions (decompositions) of natural numbers.

gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2, equal to the number of decays in which each part is equal to 1 or 4 mod 5 is.

gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2 and in which the smallest part is greater than or equal to 2 is equal the number of decays whose parts are equal to 2 or 3 mod 5.

This will be illustrated as examples in the following two tables: The following continued fraction

creates a quotient of module functions and it also makes these shown continued fractions modular: This definition applies[5] for the continued fraction mentioned: This is the definition of the Ramanujan theta function: With this function, the continued fraction R can be created this way: The connection between the continued fraction and the Rogers–Ramanujan functions was already found by Rogers in 1894 (and later independently by Ramanujan).

has the following identities to the remaining Rogers–Ramanujan functions and to the Ramanujan theta function described above: The following definitions are valid for the Jacobi "Theta-Nullwert" functions: And the following product definitions are identical to the total definitions mentioned: These three so-called theta zero value functions are linked to each other using the Jacobian identity: The mathematicians Edmund Taylor Whittaker and George Neville Watson[7][8][9] discovered these definitional identities.

The Legendre's elliptic modulus is the numerical eccentricity of the corresponding ellipse.

For the Rogers–Ramanujan continued fraction R(q) this formula is valid based on the described modular modifications of G and H: These functions have the following values for the reciprocal of Gelfond's constant and for the square of this reciprocal: The Rogers–Ramanujan continued fraction takes the following ordinate values for these abscissa values:

in this already mentioned way: The Dedekind eta function identities for the functions G and H result by combining only the following two equation chains: The quotient is the Rogers Ramanujan continued fraction accurately: But the product leads to a simplified combination of Pochhammer operators: The geometric mean of these two equation chains directly lead to following expressions in dependence of the Dedekind eta function in their Weber form: In this way the modulated functions

are represented directly using only the continued fraction R and the Dedekind eta function quotient!

With the Pochhammer products alone, the following identity then applies to the non-modulated functions G and H: For the Dedekind eta function according to Weber's definition[10] these formulas apply: The fourth formula describes the pentagonal number theorem[11] because of the exponents!

with all associated number partitions are listed in the following table: The following further simplification for the modulated functions

This connection applies especially to the Dedekind eta function from the fifth power of the elliptic nome: These two identities with respect to the Rogers–Ramanujan continued fraction were given for the modulated functions

But along with the Abel–Ruffini theorem this function in relation to the eccentricity can not be represented by elementary expressions.

Third example: Fourth example: For that function, a further expression is valid: In this way the accurate eccentricity dependent formulas for the functions G and H can be generated: Following Dedekind eta function quotient has this eccentricity dependency: This is the eccentricity dependent formula for the continued fraction R: The last three now mentioned formulas will be inserted into the final formulas mentioned in the section above:

The general case of quintic equations in the Bring–Jerrard form has a non-elementary solution based on the Abel–Ruffini theorem and will now be explained using the elliptic nome of the corresponding modulus, described by the lemniscate elliptic functions in a simplified way.

can be determined as follows: Alternatively, the same solution can be presented in this way: The mathematician Charles Hermite determined the value of the elliptic modulus k in relation to the coefficient of the absolute term of the Bring–Jerrard form.

In his essay "Sur la résolution de l'Équation du cinquiéme degré Comptes rendus" he described the calculation method for the elliptic modulus in terms of the absolute term.

The Italian version of his essay "Sulla risoluzione delle equazioni del quinto grado" contains exactly on page 258 the upper Bring–Jerrard equation formula, which can be solved directly with the functions based on the corresponding elliptic modulus.

This corresponding elliptic modulus can be worked out by using the square of the Hyperbolic lemniscate cotangent.

For the derivation of this, please see the Wikipedia article lemniscate elliptic functions!

The elliptic nome of this corresponding modulus is represented here with the letter Q: The abbreviation ctlh expresses the Hyperbolic Lemniscate Cotangent and the abbreviation aclh represents the Hyperbolic Lemniscate Areacosine!

Two examples of this solution algorithm are now mentioned: First calculation example: Quintic Bring–Jerrard equation: Solution formula: Decimal places of the nome: Decimal places of the solution: Second calculation example: Quintic Bring–Jerrard equation: Solution: Decimal places of the nome: Decimal places of the solution: The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics.

The demodularized standard form of the Ramanujan's continued fraction unanchored from the modular form is as follows:: James Lepowsky and Robert Lee Wilson were the first to prove Rogers–Ramanujan identities using completely representation-theoretic techniques.

They proved these identities using level 3 modules for the affine Lie algebra

Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels.

First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for the affine Lie algebra