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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An elliptic $BC_n$ Bailey Lemma, multiple Rogers–Ramanujan identities and Euler’s Pentagonal Number Theorems


Author: Hasan Coskun
Journal: Trans. Amer. Math. Soc. 360 (2008), 5397-5433
MSC (2000): Primary 05A19, 11B65; Secondary 05E20, 33D67
DOI: https://doi.org/10.1090/S0002-9947-08-04457-7
Published electronically: April 17, 2008
MathSciNet review: 2415079
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Abstract: 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 system $BC_n$ are proved as applications, including a $_6\varphi _5$ summation formula, a generalized Watson transformation and an unspecialized Rogers–Selberg identity. The last identity is specialized to give an infinite family of multilateral Rogers–Selberg identities. Standard determinant evaluations are then used to compute $B_n$ and $D_n$ generalizations of the Rogers–Ramanujan identities in terms of determinants of theta functions. Starting with the $BC_n$ $_6\varphi _5$ summation formula, a similar program is followed to prove an infinite family of $D_n$ Euler Pentagonal Number Theorems.


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Additional Information

Hasan Coskun
Affiliation: Department of Mathematics, Binnion Hall, Room 314, Texas A&M University–Com- merce, Commerce, Texas 75429
Email: hasan\_coskun@tamu-commerce.edu

Keywords: Elliptic Bailey Lemma, multiple Rogers–Ramanujan identities, multiple Euler’s Pentagonal Number Theorems, affine root systems, determinant evaluations, theta functions, Macdonald identities
Received by editor(s): May 22, 2006
Received by editor(s) in revised form: August 9, 2006, and October 16, 2006
Published electronically: April 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.