David Bressoud, Mourad Ismail, Dennis Stanton
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 supersymmetric analogues of Virasoro characters.
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Recently, Garvan obtained two-variable Hecke-Rogers identities for three universal mock theta functions $g_2(z;q),\,g_3(z;q),\,K(z;q)$ by using basic hypergeometric functions, and he proposed a problem of finding direct proofs of these identities by using Bailey pair technology. In this paper, we give proofs of Garvan's identities by applying Bailey's transform with the conjugate Bailey pair of Warnaar and three Bailey pairs deduced from two special cases of $_6ψ_6$ given by Slater. In particular, we obtain a compact form of two-variable Hecke-Rogers identity related to $g_3(z;q)$, which imply…
Bailey pairs and quantum $q$-series identites. I. The classical identities
本文利用 Bailey 对及换基公式证明单位根处的经典 $q$-级数恒等式，并通过 Bailey 链将其推广至无穷族。
Bilateral Bailey Lattices and Andrews-Gordon Type Identities
本文利用简单引理将Bailey格推广至双边版本，并以此为基础，给出了多种Bailey格的双边版本与简单证明，推导出多个恒等式的新$m$-版本，并提供了Bressoud一般恒等式的新初等证明。
Asymmetric bilateral Bailey pairs and Rogers-Ramanujan type identities
本文通过非对称双侧Bailey对推导出罗杰斯-拉马努金型、安德鲁斯-戈登型及假θ函数恒等式，并基于Dousse等人构建的Bailey晶格得到非对称双侧Bailey引理，进一步推导出Appell-Lerch级数恒等式。
Bilateral Bailey pairs and Rogers-Ramanujan type identities
本文利用q-二项式定理发现双边Bailey对，结合双边Bailey引理、链和格，统一推导出多种Rogers-Ramanujan型恒等式及Appell-Lerch、Andrews-Gordon和Hecke型级数恒等式。
Two new Bailey pairs and their $q$-identities of Rogers-Ramanujan type modulo 15, 24, and 30
该研究通过推广 Euler 五边形数定理构造两个新 Bailey 对，并结合 Bailey 引理得到模 15、24、30 的 Rogers–Ramanujan 型 $q$-恒等式与级数变换。
We rewrite the recently constructed q-hypergeometric integral Bailey pair in a general form. Then with the help of the Bailey pair and $q$-beta hypergeometric sum-integral, we construct the star-triangle relation.
We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.
Bailey Pairs and an Identity of Chern-Li-Stanton-Xue-Yee
本文利用 Bailey 对方法简洁证明了 Chern-Li-Stanton-Xue-Yee 恒等式，并推广得到相关恒等式及假 theta 伴随式。
It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of $q$. By inserting Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers--Ramanujan type identities and also results related to mock theta functions and Hecke--type series. In this paper, by constructing a new Bailey pair involving Chebyshev pol…
In this paper it is shown that the one-dimensional configuration sums of the solvable lattice models of Andrews, Baxter and Forrester and the string functions associated with admissible representations of the affine Lie algebra A$_1^{(1)}$ as introduced by Kac and Wakimoto can be exploited to yield a very general class of conjugate Bailey pairs. Using the recently established fermionic or constant-sign expressions for the one-dimensional configuration sums, our result is employed to derive fermionic expressions for fractional-level string functions, parafermion characters and A$_1^{(1)}$ branc…
This is a paper published in 2001 based on a talk given in 1999 celebrating the 50th anniversary of W. N. Bailey's influential q-series paper "Identities of the Rogers-Ramanujan type". In no more than 13 pages I give a brief but reasonably complete overview of the mathematics related to Bailey's lemma.
Andrews--Gordon and Stanton type identities: bijective and Bailey lemma approaches
本文利用双射方法、Bailey 引理和 Bailey 格为 Stanton 提出的 Andrews--Gordon 与 Bressoud 恒等式的两类推广给出新证明，并由此自然导出若干经典恒等式的新 Stanton 型推广。
We give an easy approach to L. Slater's Bailey pairs A(1)-A(8) with the help of q-Lucas polynomials.
Chu has recently shown that the Abel lemma on summations by parts can serve as the underlying relation for Bailey's ${}_6ψ_6$ bilateral summation formula. In other words, the Abel lemma spells out the telescoping nature of the ${}_6ψ_6$ sum. We present a systematic approach to compute Abel pairs for bilateral and unilateral basic hypergeometric summation formulas by using the $q$-Gosper algorithm. It is demonstrated that Abel pairs can be derived from Gosper pairs. This approach applies to many classical summation formulas.
By utilizing a two-dimensional extension of a very general series transform given by Bailey, Exton [Indian J. pure appl. Math. 24 (6) (1993), 401-408] deduced a very general double generating relation of a product of a pair of Laguerre polynomials and obtained a number of useful relations with elementary functions, Bessel functions, Hermite polynomials and single series expansions of pairs of Laguerre polynomials. Unfortunately, some of the results given by Exon contain errors and thus this is the aim of this short note to provide the corrected form of these results.
In this paper, we study the interaction of an antisense RNA and its target mRNA, based on the model introduced by Alkan {\it et al.} (Alkan {\it et al.}, J. Comput. Biol., Vol:267--282, 2006). Our main results are the derivation of the partition function \cite{Backhofen} (Chitsaz {\it et al.}, Bioinformatics, to appear, 2009), based on the concept of tight-structure and the computation of the base pairing probabilities. This paper contains the folding algorithm {\sf rip} which computes the partition function as well as the base pairing probabilities in $O(N^4M^2)+O(N^2M^4)$ time and $O(N^2M^2)…
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