In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Rogers–Ramanujan identities are two identities related to
basic hypergeometric series
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series.
A series ''x'n'' is called h ...
and
integer partitions
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same parti ...
. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by
Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis ...
some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof . independently rediscovered and proved the identities.
Definition
The Rogers–Ramanujan identities are
:
and
:
.
Here,
denotes the
q-Pochhammer symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer symb ...
.
Combinatorial interpretation
Consider the following:
*
is the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for partitions with exactly
parts such that adjacent parts have difference at least 2.
*
is the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for partitions such that each part is
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to either 1 or 4
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
5.
*
is the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for partitions with exactly
parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2.
*
is the
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
for partitions such that each part is
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
to either 2 or 3
modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is t ...
5.
The Rogers–Ramanujan identities could be now interpreted in the following way. Let
be a non-negative integer.
# The number of partitions of
such that the adjacent parts differ by at least 2 is the same as the number of partitions of
such that each part is congruent to either 1 or 4 modulo 5.
# The number of partitions of
such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of
such that each part is congruent to either 2 or 3 modulo 5.
Alternatively,
# The number of partitions of
such that with
parts the smallest part is at least
is the same as the number of partitions of
such that each part is congruent to either 1 or 4 modulo 5.
# The number of partitions of
such that with
parts the smallest part is at least
is the same as the number of partitions of
such that each part is congruent to either 2 or 3 modulo 5.
Modular functions
If ''q'' = e
2πiτ, then ''q''
−1/60''G''(''q'') and ''q''
11/60''H''(''q'') are
modular function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
s of τ.
Applications
The Rogers–Ramanujan identities appeared in Baxter's solution of the
hard hexagon model In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjacent.
The model was solved by , who found that i ...
in statistical mechanics.
Ramanujan's continued fraction is
:
Relations to affine Lie algebras and vertex operator algebras
James Lepowsky
James "Jim" Lepowsky (born July 5, 1944, in New York City) is a professor of mathematics at Rutgers University, New Jersey. Previously he taught at Yale University. He received his Ph.D. from M.I.T. in 1970 where his advisors were Bertram Kostant ...
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
. In the course of this proof they invented and used what they called
-algebras.
Lepowsky and Wilson's approach is universal, in that it is able to treat all
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody a ...
s 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
.
See also
*
Rogers polynomials
In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identitie ...
*
Continuous q-Hermite polynomials
In mathematics, the continuous ''q''-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic h ...
References
*
*
*
*
*
*
W.N. Bailey, ''Generalized Hypergeometric Series'', (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge.
* George Gasper and Mizan Rahman, ''Basic Hypergeometric Series, 2nd Edition'', (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. .
*
Bruce C. Berndt
Bruce Carl Berndt (born March 13, 1939, in St. Joseph, Michigan)
is an American mathematician. Berndt attended college at Albion College, graduating in 1961, where he also ran track. He received his master's and doctoral degrees from the Universi ...
, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son,
The Rogers-Ramanujan Continued Fraction', J. Comput. Appl. Math. 105 (1999), pp. 9–24.
* Cilanne Boulet,
Igor Pak Igor Pak (russian: link=no, Игорь Пак) (born 1971, Moscow, Soviet Union) is a professor of mathematics at the University of California, Los Angeles, working in combinatorics and discrete probability. He formerly taught at the Massachusetts ...
,
A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities', Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030.
*
* James Lepowsky and Robert L. Wilson, ''Construction of the affine Lie algebra
, Comm. Math. Phys. 62 (1978) 43-53.
* James Lepowsky and Robert L. Wilson, ''A new family of algebras underlying the Rogers-Ramanujan identities'', Proc. Natl. Acad. Sci. USA 78 (1981), 7254-7258.
* James Lepowsky and Robert L. Wilson, ''The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities'', Invent. Math. 77 (1984), 199-290.
* James Lepowsky and Robert L. Wilson, ''The structure of standard modules, II: The case
, principal gradation'', Invent. Math. 79 (1985), 417-442.
* Stefano Capparelli, ''Vertex operator relations for affine algebras and combinatorial identities'', Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. 1988. 107 pp.
External links
*
*
{{DEFAULTSORT:Rogers-Ramanujan identities
Hypergeometric functions
Integer partitions
Mathematical identities
Q-analogs
Modular forms
Srinivasa Ramanujan