On identities of the Rogers-Ramanujan type

A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix. 2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13     

A Higher Level Bailey Lemma: Proof and Application

In a recent letter, new representations were proposed for the pair of sequences (,), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs (,) labelled by the Lie algebra AN – 1, two nonnegative integers and k and a partition , whose parts do not exceed N – 1. Our results give rise to what we call a higher level Bailey lemma. As an application it is shown how this lemma can be applied to yield general q-series identities, which generalize some well-known results of Andrews and Bressoud.     

An A Bailey lemma and Rogers-Ramanujan-type identities

Using new -functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A version of the classical Bailey lemma. We apply our result, which is distinct from the A Bailey lemma of Milne and Lilly, to derive Rogers-Ramanujan-type identities for characters of the W algebra.

     

A Bailey Tree for Integrals

We introduce the notion of integral Bailey pairs. Using the single-variable elliptic beta integral, we construct an infinite binary tree of identities for elliptic hypergeometric integrals. Two particular sequences of identities are described explicitly.     

Identities of the Rogers-Ramanujan-Bailey type

A multiparameter generalization of the Bailey pair is defined in such a way as to include as special cases all Bailey pairs considered by W.N. Bailey in his paper [Identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 50 (1949) 421-435]. This leads to the derivation of a number of elegant new Rogers-Ramanujan type identities.     

普通型Bailey引理和 Riordan链

本文主要研究在超几何函数和Ramanujan-Rogers类型恒等式有非常重要作用的.Beilay引理和Riordan链的关系,证明了普通型Bailey引理本质上是一个特殊Riordan群的Riordan链.     

New finite Rogers-Ramanujan identities

We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson’s transformation formula by specialization or through Bailey’s method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.      

拉格朗日反演的q模拟与Bailey引理的注记

本文主要揭示了Gessel Ira.等给出的拉格朗日反演的q—模拟形式与An-drews G.E.的Bailey引理之间的相互转化的联系,做为例证,给出了利用这些关系得到的古典超几何级数(hypergeometric series)变换和求和公式的新证明,同时得到了模5、7、9、27四个新的Roger’s-Ramanujan类型的恒等式,其具有十分重要的组合意义。     

Extensions of the well-poised and elliptic well-poised Bailey lemma

We establish a number of extensions of the well-poised Bailey lemma and elliptic well-poised Bailey lemma. As application we prove some new transformation formulae for basic and elliptic hyper-geometric series, and embed some recent identities of Andrews, Berkovich and Spiridonov in a well-poised Bailey tree.     

On a characteristic equation of well-poised Bailey chains

In this paper, we find by inverse technique two solutions of a system of linear equations which together serve as a sufficient and necessary condition for well-poised Bailey chains. Using these two solutions, we establish a new well-poised Bailey chain, two usual Bailey chains, and a well-poised extension of Bailey’s lemma. Their applications to q-series are also investigated. X. Ma was supported by Natural Science Foundation of China (No. 10771156).     

Ramanujan-Slater type identities related to the moduli 18 and 24

We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.     

一些Rogers-Ramanujan恒等式的广义形式

本文用简便的方法，得到了一些Ｒｏｇｅｒｓ－Ｒａｍａｎｕｊａｎ恒等式的广义形式．     

Bailey pairs and indefinite quadratic forms

We construct classes of Bailey pairs where the exponent of q   in _αn${\alpha }_n$ is an indefinite quadratic form. As an application we obtain families of q-hypergeometric mock theta multisums.     

The numerical range of a tridiagonal operator

Let r be a real number and A a tridiagonal operator defined by Aej=ej−1+rjej+1, j=1,2,…, where {e₁,e₂,…} is the standard orthonormal basis for ?²(N). Such tridiagonal operators arise in Rogers-Ramanujan identities. In this paper, we study the numerical ranges of these tridiagonal operators and finite-dimensional tridiagonal matrices. In particular, when r=−1, the numerical range of the finite-dimensional tridiagonal matrix is the convex hull of two explicit ellipses. Applying the result, we obtain that the numerical range of the tridiagonal operator is the square

     

Andrews–Gordon type identities from combinations of Virasoro characters

For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ _1,s ^p,p′+q ^Δ χ _p−1,s ^p,p′ of Virasoro (W ₂) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new. The case (p,p′)=(3,14) is particularly interesting because it relates not only to W ₂, but also to W ₃ characters, and offers W ₃ analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al. which involve Gaussian polynomials. BF is partially supported by grant number RFBR 05-01-01007, and OF by the Australian Research Council (ARC).     

New proofs of identities of Lebesgue and Göllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities:

     

The method of combinatorial telescoping

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial derivation of Watson's identity, which implies the Rogers-Ramanujan identities.     

Definable Sets in Mann Pairs

Consider structures (Ω, k, Γ) where Ω is an algebraically closed field of characteristic zero, k is a subfield, and Γ is a subgroup of the multiplicative group of Ω. Certain pairs (k, Γ) have been singled out as Mann pairs in [4 van den Dries , L. , Günayd?n , A. ( 2010 ). Mann pairs . Trans. Amer. Math. Soc. 362 : 2393 – 2414 . [Google Scholar]]. We give new examples of such Mann pairs, we axiomatize for each Mann pair (k, Γ) the first-order theory of (Ω, k, Γ) in a cleaner way than in [4 van den Dries , L. , Günayd?n , A. ( 2010 ). Mann pairs . Trans. Amer. Math. Soc. 362 : 2393 – 2414 . [Google Scholar]], and, as the main result of the article, we characterize the subsets of Ω ⁿ that are definable in (Ω, k, Γ).     

Skew Pairs of Idempotents in Transformation Semigroups

An ordered pair （e, f） of idempotents of a regular semigroup is called a skew pair if ef is not idempotent whereas fe is idempotent. We have shown previously that there are four distinct types of skew pairs of idempotents. Here we investigate the ubiquity of such skew pairs in full transformation semigroups.     

Lifting Bailey pairs to WP-Bailey pairs

A pair of sequences such that and