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

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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.## References

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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
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2415079