Bailey Lemma Research Articles

On the q-binomial identities involving the Legendre symbol modulo 3

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

Bm$(1-xy,y-x)$-expansion formula and its applications

In this paper, we first show a $(1-xy,y-x)$-expansion formula for an arbitrary analytic function $F(x)$ with respect to the basis <disp-formula><tex-math><![CDATA[$$\bigg\{\prod_{k=0}^{n-1}\frac{x-b_k}{1-x_kx}\,\bigg|\, n\geq 0\bigg\},$$]]></tex-math></disp-formula> which is mainly based on the $(f,g)$-inversion formula. Subsequently, by specializing $F(x)$, $x_n$ and $b_n$, we not only find the corresponding proofs of many classical results such as the Rogers-Fine identity, Andrews' four parametric reciprocal theorem, and Ramanujan's ${~}_1\psi_1$ summation formula, but also construct a lot of $q$-series transformations and summation formulas, including a generalization of Andrews' WP Bailey lemma.

ANDREWS’ TYPE WP-BAILEY LEMMA AND ITS APPLICATIONS

ver the years, the study of Bailey transform, Bailey lemma, Bailey pair, their variants and their applications are the major subjects of interest. Of course, it is due to the efficiency of the Bailey transform and lemma in producing many ordinary and q-hypergeometric identities, multiple series summation and transformation formulas, and the Rogers-Ramanujan type identities. Andrews investigated a WP-Bailey lemma and the pairs with the help of Bailey transform and used it to derive well-known summations and multiple series transformations. In this research paper, we investigate an Andrews’ type WP-Bailey lemma and the pairs with the help of First Bailey Type Transform due to Joshi and Vyas. The investigated Andrews’ type WP-Bailey lemma is then applied to obtain terminating multiple q-hypergeometric identities and construct the WP-Bailey type chains and a binary tree. The paper is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including number theory, theory of partitions and combinatorial analysis as well as in the study of combinatorial generating functions.

New infinite hierarchies of polynomial identities related to the Capparelli partition theorems

We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the q↦1/q duality transformation of the base identities and some related partition theoretic relations.

The $2$-fold Bailey lemma and mock theta functions

In this paper, we obtain two identities related to mock theta functions ℱ2(q) and ϕ(q) by employing the 2-fold Bailey lemma. And, we also deduce an identity as mock theta function ν(q) by using Bailey’s lemma.

Some implications of the $2$-fold Bailey lemma

The $2$-fold Bailey lemma is a special case of the $s$-fold Bailey lemma introduced by Andrews in 2000. We examine this special case and its applications to partitions and recently discovered $q$-series identities. Our work provides a general comparison of the utility of the $2$-fold Bailey lemma and the more widely applied $1$-fold Bailey lemma. We also offer a discussion of the $spt_M(n)$ function and related identities.

On the Andrews–Yee Identities Associated with Mock Theta Functions

In this paper, we generalize the Andrews–Yee identities associated with the third-order mock theta functions $$\omega (q)$$ and $$\nu (q)$$. We obtain some q-series transformation formulas, one of which gives a new Bailey pair. Using the classical Bailey lemma, we derive a product formula for two $${}_2\phi _1$$ series. We also establish recurrence relations and transformation formulas for two finite sums arising from the Andrews–Yee identities.

The rarefied elliptic Bailey lemma and the Yang–Baxter equation

An elliptic Bailey lemma is formulated on the basis of the univariate rarefied elliptic beta integral. It leads to a generalized operator star-triangle relation and a new solution of the Yang–Baxter equation written as an integral operator with a rarefied elliptic hypergeometric kernel.

Polynomial identities implying Capparelli's partition theorems

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving q-trinomial coefficients. We follow Kurşungöz's ideas to provide direct combinatorial interpretations of some of our expressions. We make use of the trinomial analogue of Bailey's lemma to derive new identities. These identities relate certain triple sums and products. A couple of new Slater type identities involving bases q2, q3, q6, and q12 are also proven. We also discuss a new infinite hierarchy containing these Slater type identities.

An overpartition analogue of the Andrews-Göllnitz-Gordon theorem

In 1967, Andrews found a combinatorial generalization of the Göllnitz-Gordon theorem, which can be called the Andrews-Göllnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-Göllnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-Göllnitz-Gordon theorem for i=k. In this paper, we give an overpartition analogue of this theorem for k≥i≥1. By using Bailey's lemma and a change of base formula due to Bressoud, Ismail and Stanton, we obtain an overpartition analogue of Bressoud's identity. We then give a combinatorial interpretation of this identity by introducing the Göllnitz-Gordon marking of an overpartition, which yields an overpartition analogue of the Andrews-Göllnitz-Gordon theorem.

Matrix Bailey Lemma and the Star-Triangle Relation

We compare previously found finite-dimensional matrix and integral operator realizations of the Bailey lemma employing univariate elliptic hypergeometric functions. With the help of residue calculus we explicitly show how the integral Bailey lemma can be reduced to its matrix version. As a consequence, we demonstrate that the matrix Bailey lemma can be interpreted as a star-triangle relation, or as a Coxeter relation for a permutation group.

4d mathcal{N} = 1 quiver gauge theories and the An Bailey lemma

We study the integral Bailey lemma associated with the An-root system and identities for elliptic hypergeometric integrals generated thereby. Interpreting integrals as superconformal indices of four-dimensional mathcal{N} = 1 quiver gauge theories with the gauge groups being products of SU(n + 1), we provide evidence for various new dualities. Further confirmation is achieved by explicitly checking that the ‘t Hooft anomaly matching conditions holds. We discuss a flavour symmetry breaking phenomenon for supersymmetric quantum chromodynamics (SQCD), and by making use of the Bailey lemma we indicate its manifestation in a web of linear quivers dual to SQCD that exhibits full s-confinement.

A duality web of linear quivers

We show that applying the Bailey lemma to elliptic hypergeometric integrals on the $A_n$ root system leads to a large web of dualities for $\mathcal{N} = 1$ supersymmetric linear quiver theories. The superconformal index of Seiberg's SQCD with $SU(N_c)$ gauge group and $SU(N_f)\times SU(N_f)\times U(1)$ flavour symmetry is equal to that of $N_f-N_c-1$ distinct linear quivers. Seiberg duality further enlarges this web by adding new quivers. In particular, both interacting electric and magnetic theories with arbitrary $N_c$ and $N_f$ can be constructed by quivering an $s$-confining theory with $N_f=N_c+1$.

The SU(r)2 string functions as q-diagrams

A generalized Roger Ramanujan (GRR) type expression for the characters of A-type parafermions has been a long standing puzzle dating back to conjectures made regarding some of the characters in the 90s. Not long ago we have put forward such GRR type identities describing any of the level two ADE-type generalized parafermions characters at any rank. These characters are the string functions of simply laced Lie algebras at level two, as such, they are also of mathematical interest. In our last joint paper we presented the complete derivation for the D-type generalized parafermions characters identities. Here we generalize our previous discussion and prove the GRR type expressions for the characters of A-type generalized parafermions. To prove the A-type GRR conjecture we study further the q-diagrams, introduced in our last joint paper, and examine the diagrammatic interpretations of known identities among them Slater identities for the characters of the first minimal model, which is the Ising model, and the Bailey lemma.

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.

A multilateral Bailey lemma and multiple Andrews–Gordon identities

A multilateral Bailey lemma is proved, and multiple analogues of the Rogers–Ramanujan identities and Euler’s pentagonal theorem are constructed as applications. The extreme cases of the Andrews–Gordon identities are also generalized using the multilateral Bailey lemma where their final form are written in terms of determinants of theta functions.

Shifted versions of the Bailey and Well-Poised Bailey lemmas

The Bailey lemma is a famous tool to prove Rogers–Ramanujan type identities. We use shifted versions of the Bailey lemma to derive m-versions of multisum Rogers–Ramanujan type identities. We also apply this method to the Well-Poised Bailey lemma and obtain a new extension of the Rogers–Ramanujan identities.

BILATERAL BAILEY LEMMA AND FALSE THETA FUNCTIONS

By examining the transformation formulae between unilateral series and bilateral ones derived from the bilateral Bailey lemma, we establish numerous identities of false theta functions, including most of the known ones discovered mainly by Rogers [40] and Ramanujan in his Lost Notebook [39].

Bilateral Bailey lemma and Rogers–Ramanujan identities

We prove a bilateral Bailey lemma and establish twenty-five transformation formulae between unilateral and bilateral summations for nonterminating basic hypergeometric series. Then they are employed to show numerous identities of Rogers–Ramanujan type, including most of Slater's collection of 130 identities.

Positivity preserving transformations for 𝑞-binomial coefficients

Several new transformations forqq-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The newqq-binomial transformations are also applied to obtain multisum Rogers–Ramanujan identities, to find new representations for the Rogers–Szegö polynomials, and to make some progress on Bressoud’s generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.