Generalized Rogers Ramanujan Research Articles

Generalized Rogers–Ramanujan expressions for some non-singlet twisted affine algebras

Hatayama et al. described generalized Rogers–Ramanujan (GRR) expressions for the string functions of the singlet representation of twisted affine algebras. We give here such GRR expressions for some non-singlet string functions. In the case of the algebra [Formula: see text], this gives all the string functions. We verify these expressions using Freudenthal–Kac formula.

Overpartition analogues for the generalized Rogers–Ramanujan identities of Bressoud

In the past two decades, there have been a lot of research centred around overpartitions, some of which concern overpartition analogues of Rogers–Ramanujan type identities. In this paper, we present Rogers–Ramanujan type overpartition identities by considering Bressoud’s even moduli generalization of the Rogers–Ramanujan identity and its overpartition analogue of Chen, Sang and Shi given in 2015. We first introduce another overpartition function C¯k,a(n) and show that C¯k,a(n) equals the overpartition function B¯k,a,0(n) of Chen, Sang and Shi. Next, we study parity constrains on parts of overpartitions. Recently, Sang, Shi and Yee obtained Rogers–Ramanujan type identities for overpartitions by adding some parity constraints to even or odd parts of overpartitions Chen, Sang and Shi introduced in 2013. We make some modifications and add constraints to even or odd parts of overpartitions counted by C¯k,a(n) and B¯k,a,0(n) obtaining further Rogers–Ramanujan type overpartition identities.

Generalized Rogers–Ramanujan identities for twisted affine algebras

The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

Generalized Rogers Ramanujan Identities Motivated by AGT Correspondence

AGT correspondence and its generalizations attracted a great deal of attention recently. In particular it was suggested that $U(r)$ instantons on $R^4/Z_p$ describe the conformal blocks of the coset ${\cal A}(r,p)=U(1)\times sl(p)_r\times {sl(r)_p\times sl(r)_n\over sl(r)_{n+p}}$, where $n$ is a parameter. Our purpose here is to describe Generalized Rogers Ramanujan (GRR) identities for these cosets, which expresses the characters as certain $q$ series. We propose that such identities exist for the coset ${\cal A}(r,p)$ for all positive integers $n$ and all $r$ and $p$. We treat here the case of $n=1$ and $r=2$, finding GRR identities for all the characters.

Anti-lecture hall compositions and overpartitions

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k − 2 equals the number of overpartitions of n with non-overlined parts not congruent to 0 , ± 1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers–Ramanujan type identities for overpartitions which are analogous to the Rogers–Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers–Ramanujan identity also due to Andrews.

Zeros of generalized Rogers–Ramanujan series: Asymptotic and combinatorial properties

In this paper we study the properties of coefficients appearing in the series expansions for zeros of generalized Rogers–Ramanujan series. Our primary purpose is to address several conjectures made by M.E.H. Ismail and C. Zhang. We prove that the coefficients in the series expansion of each zero approach rational multiples of π and π 2 as q → 1 − . We also show that certain polynomials arising in connection with the zeros of Rogers–Ramanujan series generalize the coefficients appearing in the Taylor expansion of the tangent function. These polynomials provide an enumeration for alternating permutations different from that given by the classical q -tangent numbers. We conclude the paper with a method for inverting an elliptic integral associated with the zeros of generalized Rogers–Ramanujan series. Our calculations provide an efficient algorithm for the computation of series expansions for zeros of generalized Rogers–Ramanujan series.

Hadamard products for generalized Rogers–Ramanujan series

The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.

ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES

Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

Certain Bibasic Hypergeometric Transformation Formulae and Their Application to Rogers–Ramanujan Identities

In this paper, a variety of generalized bibasic hypergeometric transformation formulae have been investigated with the objective of finding a multiple series generalized Rogers–Ramanujan type of identity. It has been shown that some very interesting new multiple series identities of the Rogers–Ramanujan type can be found where there is more than one infinite product in which the terms in each product advance in different powers ofq.

PATH SPACES AND $\mathcal W$ FUSION IN MINIMAL MODELS

Product forms of characters of Virasoro minimal models are obtained which factorize into (2, odd) × (3, even) characters. These are related by generalized Rogers–Ramanujan identities to sum forms, allowing for a quasiparticle interpretation. The corresponding dilogarithm identities are given and the factorization is used to analyze the related path space structure as well as the fusion of the maximally extended chiral algebra.