Mathematics – Combinatorics
Scientific paper
2006-05-24
Mathematics
Combinatorics
V2: 36 pages; to appear in AMS Trans; references added; typos corrected
Scientific paper
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 $_6\phi_5$ 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 $BC_n$ $_6\phi_5$ summation formula, a similar program is followed to prove an infinite family of $D_n$ Euler's Pentagonal Number Theorems.
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