Rogers-Ramanujan Identities Research Articles

A new family of algebras underlying the Rogers-Ramanujan identities and generalizations

The classical Rogers-Ramanujan identities have been interpreted by Lepowsky-Milne and the present authors in terms of the representation theory of the Euclidean Kac-Moody Lie algebra A(1) ((1)). Also, the present authors have introduced certain "vertex" differential operators providing a construction of A(1) ((1)) on its basic module, and Kac, Kazhdan, and we have generalized this construction to a general class of Euclidean Lie algebras. Starting from this viewpoint, we now introduce certain new algebras [unk](v) which centralize the action of the principal Heisenberg subalgebra of an arbitrary Euclidean Lie algebra [unk] on a highest weight [unk]-module V. We state a general (tautological) Rogers-Ramanujan-type identity, which by our earlier theorem includes the classical identities, and we show that [unk](v) can be used to reformulate the general identity. For [unk] = A(1) ((1)), we develop the representation theory of [unk](v) in considerable detail, allowing us to prove our earlier conjecture that our general Rogers-Ramanujan-type identity includes certain identities of Gordon, Andrews, and Bressoud. In the process, we construct explicit bases of all of the standard and Verma modules of nonzero level for A(1) ((1)), with an explicit realization of A(1) ((1)) as operators in each case. The differential operator constructions mentioned above correspond to the trivial case [unk](v) = (1) of the present theory.

The Rogers-Ramanujan identities: Lie theoretic interpretation and proof

The two Rogers-Ramanujan identities, which equate certain infinite products with infinite sums, are among the most intriguing of the classical formal power series identitites. It has been found by Lepowsky and Milne that the product side of each of them differs by a certain factor from the principally specialized character of a certain standard module for the Euclidean Kac-Moody Lie algebra A(1) ((1)). On the other hand, the present authors have introduced an infinite-dimensional Heisenberg subalgebra [unk] of A(1) ((1)) which leads to a construction of A(1) ((1)) in terms of differential operators given by the homogeneous components of an "exponential generating function." In the present announcement, we use [unk] to formulate a natural "abstract Rogers-Ramanujan identity" for an arbitrary standard A(1) ((1))-module which turns out to coincide with the classical identities in the cases of the two corresponding standard modules. The abstract identity equates two expressions, one a product and the other a sum, for the principally specialized character of the space Omega of highest weight vectors or "vacuum states" for [unk] in the module. The construction of A(1) ((1)) leads to a concrete realization of Omega as the span of certain spaces of symmetric polynomials occurring as the homogeneous components of exponential generating functions. The summands in the Rogers-Ramanujan identities turn out to "count" the dimensions of these spaces. For general standard A(1) ((1))-modules, we conjecture that the abstract identities agree with generalizations of the Rogers-Ramanujan identities due to Gordon, Andrews, and Bressoud.

A correspondence between partitions related to generalizations of the Rogers–Ramanujan identities

A 1-??? correspondence is established between partitions of a positive integer n of the form ( p 1 p 2 p 3… p n ) where p 1− p j+ k-1 ⩾2 and at most i−1 of the p j and partitions of n whose book differences d j all lie in the range −( i-2)⩽2 k- i-1. The same correspondence is also applied to partitions whose book differences lie in the range −( i-2)⩽ d jk ⩽2 k- i-2, and leads to further restrictions on the first set, and in particular to a correspondence between the partitions of a number n into distinct partsand the partitions of n whose book differences are all 0,1 or 2.

Two further Ramanujan pairs

AbstractIn a recent article, George E. Andrews considers a generalization of the Rogers-Ramanujan identities involving a pair of infinite sequences of positive integers, which he calls a Ramanujan pair. He lists the known Ramanujan pairs and conjectures that there are no more. The object of this note is to establish the existence of two further Ramanujan pairs.

AN ANALYTIC GENERALIZATION OF THE ROGERS-RAMANUJAN IDENTITIES WITH INTERPRETATION

Journal Article AN ANALYTIC GENERALIZATION OF THE ROGERS-RAMANUJAN IDENTITIES WITH INTERPRETATION Get access DAVID M. BRESSOUD DAVID M. BRESSOUD The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802 U.S.A. Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 31, Issue 4, December 1980, Pages 385–399, https://doi.org/10.1093/qmath/31.4.385 Published: 01 December 1980 Article history Received: 23 August 1978 Revision received: 14 January 1980 Published: 01 December 1980

A combinatorial proof of Schur’s 1926 partition theorem

One of the partition theorems published by I. J. Schur in 1926 is an extension of the Rogers-Ramanujan identities to partitions with minimal difference three. This theorem of Schur is proved here by establishing a one-to-one correspondence between the two types of partitions counted.

Some sieves for partition theory

A Combinatorial lemma due to Zolnowsky is applied to partition theory in a new way. Several even-odd formulas are derived. A combinatorial proof of the Jacobi triple-product identity is given. Some sieve formulas are proved which relate to the Euler pentagonal number theorem and the Rogers-Ramanujan identities and their generalizations. A formula is given for the number of partitions of n into parts not congruent to 0, ± x (mod y ).

A generalization of the Rogers-Ramanujan identities for all moduli

The Rogers-Ramanujan identities have been extended to odd moduli by B. Gordon and to moduli of the form 4 k + 2 by G. Andrews. We demonstrate and prove an extension to all even moduli, and provide a theorem which holds for all moduli.

On the Alder Polynomials and a New Generalization of the Rogers-Ramanujan Identities

In 1954, H. L. Alder obtained certain analytic generalizations of the Rogers-Ramanujan identities that involved a sequence of polynomials subsequently named the Alder polynomials. The object here is to obtain a new representation of the Alder polynomials from which a new generalization of the Rogers-Ramanujan identities is easily deduced.

Partially ordered sets and the rogers-ramanujan identities

Partially ordered sets and the rogers-ramanujan identities

On the Alder polynomials and a new generalization of the Rogers-Ramanujan identities

In 1954, H. L. Alder obtained certain analytic generalizations of the Rogers-Ramanujan identities that involved a sequence of polynomials subsequently named the Alder polynomials. The object here is to obtain a new representation of the Alder polynomials from which a new generalization of the Rogers-Ramanujan identities is easily deduced.

An Analytic Generalization of the Rogers-Ramanujan Identities for Odd Moduli

A (k - 1)-fold Eulerian series expansion is given for II(1 - q(n))(-1), where the product runs over all positive integers n that are not congruent to 0,i or - i modulo 2k + 1. The Rogers-Ramanujan identities are the cases k = i = 2 and k = i + 1 = 2.

The Rogers–Ramanujan Identities

The Rogers–Ramanujan Identities

Partially ordered sets and the rogers-ramanujan identities

Partially ordered sets and the rogers-ramanujan identities

A new property of partitions with applications to the Rogers-Ramanujan identities

The order of a partition π (relative to N) is defined as the largest i for which the number of summands in the closed interval [ i, i + N − 1] is at least i. By studying the generating function for partitions into distinct parts not exceeding 2 N with given order, we are able to derive an identity of importance in the theory of partitions.

Partition theorems related to the Rogers-Ramanujan identities

In this paper a partition theorem is proved which contains the Rogers-Ramanujan identities and Euler's partition theorem as special cases. Other partition theorems of the Rogers-Ramanujan type are proved.

A SIMPLE PROOF OF THE ROGERS-RAMANUJAN IDENTITIES

Journal Article A SIMPLE PROOF OF THE ROGERS-RAMANUJAN IDENTITIES Get access J. M. DOBBIE J. M. DOBBIE Cambridge, Mass Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 13, Issue 1, 1962, Pages 31–34, https://doi.org/10.1093/qmath/13.1.31 Published: 01 January 1962 Article history Received: 09 February 1961 Published: 01 January 1962