Related papers: A Higher-level Bailey Lemma

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…

In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…

We announce a higher-dimensional generalization of the Bailey Transform, Bailey Lemma, and iterative ``Bailey chain'' concept in the setting of basic hypergeometric series very well-poised on unitary $A_{\ell}$ or symplectic $C_{\ell}$…

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…

Rogers-Ramanujan type identities occur in various branches of mathematics and physics. As a classic and powerful tool to deal with Rogers-Ramanujan type identities, the theory of Bailey's lemma has been extensively studied and generalized.…

Using new $q$-functions recently introduced by Hatayama et al. and by (two of) the authors, we obtain an A_2 version of the classical Bailey lemma. We apply our result, which is distinct from the A_2 Bailey lemma of Milne and Lilly, to…

An elliptic $BC_n$ generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter $BC_n$ Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root…

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…

The theory of Bailey's transform provides a systematic method for deriving $q$-identities, the key factor of which is the Bailey pair. The concept of Bailey pair was first extended to bilateral version by Paule. In this paper, following…

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by…

We use Bailey pairs to prove $q$-series identities at roots of unity due to Cohen and Bryson-Ono-Pitman-Rhoades. The proofs use Bailey pairs with quadratic forms developed in the study of mock theta functions. In addition to the standard…

We show that the Bailey lattice can be extended to a bilateral version in just a few lines from the bilateral Bailey lemma, using a very simple lemma transforming bilateral Bailey pairs relative to $a$ into bilateral Bailey pairs relative…

Using the theory of Kostka polynomials, we prove an A_{n-1} version of Bailey's lemma at integral level. Exploiting a new, conjectural expansion for Kostka numbers, this is then generalized to fractional levels, leading to a new expression…

Versions of Bailey's lemma which change the base from q to q^2 or q^3 are given. Iterates of these versions give many new versions of multisum Rogers-Ramanujan identities. We also prove Melzer's conjectures for the Fermionic forms of the…

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…

A generalized Bailey pair, which contains several special cases considered by Bailey (\emph{Proc. London Math. Soc. (2)}, 50 (1949), 421--435), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of…

In 2018, Stanton proved two types of generalisations of the celebrated Andrews--Gordon and Bressoud identities (in their $q$-series version): one with a similar shape to the original identities, and one involving binomial coefficients. In…

We evaluate $q$-Bessel functions at an infinite sequence of points and introduce a generalization of the Ramanujan function and give an extension of the $m$-version of the Rogers-Ramanujan identities. We also prove several generating…

The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of…

In this paper, we investigate applications of the ordinary derivative operator, instead of the $q$-derivative operator, to the theory of $q$-series. As main results, many new summation and transformation formulas are established which are…