Related papers: An Elliptic $BC_n$ Bailey Lemma, Multiple Rogers--…

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…

We propose a generalization of Bailey's lemma, useful for proving $q$-series identities. As an application, generalizations of Euler's identity, the Rogers-Ramanujan identities, and the Andrews-Gordon identities are derived. This…

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…

Lucy Slater used Bailey's $_6\psi_6$ summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's…

A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6\psi_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts…

We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the…

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…

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.…

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in…

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate…

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 this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of…

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…

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 provide proofs of two ${}_5\psi_5$ summation formulas of Bailey using a ${}_5\phi_4$ identity of Carlitz. We show that in the limiting case, the two ${}_5\psi_5$ identities give rise to two ${}_3\psi_3$ summation formulas…

In 1981, Andrews gave a four-variable generalization of Ramanujan's ${_1\psi_1}$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${_8\phi_7}$ series and…

Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system $C_n$. In a…

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…

The Ramanujan $_1\psi_1$ summation theorem in studied from the perspective of $q$-Jackson integrals, $q$-difference equations and connection formulas. This is an approach which has previously been shown to yield Bailey's very-well-poised…