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In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers, and were subsequently rediscovered 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. Issai Schur independently rediscovered and proved the identities.
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. 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.
Contents
Definition
The Rogers–Ramanujan identities are
G
(
q
)
=
∑
n
=
0
∞
q
n
2
(
q
;
q
)
n
=
1
(
q
;
q
5
)
∞
(
q
4
;
q
5
)
∞
=
Combinatorial interpretation
Consider the following:
q
n
2
(
q
;
q
)
n
{\displaystyle {\frac {q^{n^{2}}}{(q;q)_{n}}}}
is the generating function for partitions with exactly
n
{\displaystyle n}
parts such that adjacent parts have difference at least 2.
Since the terms occurring in the identity are generating functions of certain partitions, the identities make statements about partitions (decompositions) of natural numbers. The number sequences resulting from the coefficients of the Maclaurin series of the Rogers–Ramanujan functions G and H are special partition number sequences of level 5:
G
(
x
)
=
1
(
x
;
x
5
)
∞
(
x
4
;
x
5
)
∞
=
1
+
∑
n
=
1
∞
P
G
(
Rogers–Ramanujan continued fractions R and S
Definition of the continued fractions
The following continued fraction
R
(
q
)
{\displaystyle R(q)}
is called Rogers–Ramanujan continued fraction, Continuing fraction
S
(
q
)
{\displaystyle S(q)}
is called alternating Rogers–Ramanujan continued fraction!
The factor
q
1
5
{\displaystyle q^{\frac {1}{5}}}
creates a quotient of module functions and it also makes these shown continued fractions modular:
This definition applies for the continued fraction mentioned:
R
(
q
)
=
q
1
/
5
Identities with Jacobi theta functions
The following definitions are valid for the Jacobi "Theta-Nullwert" functions:
An elliptic function is a modular function if this function in dependence on the elliptic nome as an internal variable function results in a function, which also results as an algebraic combination of Legendre's elliptic modulus and its complete elliptic integrals of the first kind in the K and K' form. The Legendre's elliptic modulus is the numerical eccentricity of the corresponding ellipse.
If you set
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
(where the imaginary part of
τ
∈
C
{\displaystyle \tau \in \mathbb {C} }
is positive), following two functions are modular functions!
G
M
(
q
)
=
q
−
1
60
G
(
Special values
These functions have the following values for the reciprocal of Gelfond's constant and for the square of this reciprocal:
The following further simplification for the modulated functions
G
M
{\displaystyle G_{M}}
and
H
M
{\displaystyle H_{M}}
can be undertaken. This connection applies especially to the Dedekind eta function from the fifth power of the elliptic nome:
η
W
(
q
5
)
η
W
(
q
)
=
η
W
(
q
2
)
4
η
W
(
q
)
Reduced Weber modular function
The Weber modular functions in their reduced form are an efficient way of computing the values of the Rogers–Ramanujan functions:
First of all we introduce the reduced Weber modular functions in that pattern:
w
R
n
(
ε
)
=
2
(
n
−
1
)
/
4
[
q
(
ε
)
n
;
q
(
ε
)
2
n
]
∞
[
q
Exact eccentricity identity for the functions G and H
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:
η
W
[
q
(
ε
)
2
]
η
W
[
q
(
ε
)
]
=
2
−
1
/
4
tan
[
2
arctan
(
ε
Application to quintic equations
Discovery of the corresponding modulus by Charles Hermite
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.
x
5
+
5
x
=
4
c
{\displaystyle x^{5}+5\,x=4\,c}
The real solution for all real values
c
∈
R
{\displaystyle c\in \mathbb {R} }
can be determined as follows:
x
=
S
⟨
q
{
ctlh
[
1
2
aclh
(
Calculation examples
Two examples of this solution algorithm are now mentioned:
First calculation example:
Second calculation example:
Applications in Physics
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:
Relations to affine Lie algebras and vertex operator algebras
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
s
l
2
^
{\displaystyle {\widehat {{\mathfrak {sl}}_{2}}}}
. In the course of this proof they invented and used what they called
Z
{\displaystyle Z}
-algebras. Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels. It can be used to find (and prove) new partition identities. First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for
the affine Lie algebra
A
2
(
2
)
{\displaystyle A_{2}^{(2)}}
.
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
(sequence A003114 in the OEIS)) represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 1 or 5a + 4 with a ∈
N
0
{\displaystyle \mathbb {N} _{0}}
. Thus
P
G
(
n
)
{\displaystyle P_{G}(n)}
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.
And the number sequence
P
H
(
n
)
{\displaystyle P_{H}(n)}
(sequence A003106 in the OEIS)) analogously represents the number of possibilities for the affected natural number n to decompose this number into summands of the patterns 5a + 2 or 5a + 3 with a ∈
N
0
{\displaystyle \mathbb {N} _{0}}
. Thus
P
H
(
n
)
{\displaystyle P_{H}(n)}
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 connection between the continued fraction and the Rogers–Ramanujan functions was already found by Rogers in 1894 (and later independently by Ramanujan).
The continued fraction can also be expressed by the Dedekind eta function:
The element of the fifth root can also be removed from the elliptic nome of the theta functions and transferred to the external tangent function. In this way, a formula can be created that only requires one of the three main theta functions:
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:
