An elliptic BCn Bailey Lemma, multiple Rogers-Ramanujan identities and Euler's Pentagonal Number Theorems

摘要

An elliptic BCn generalization of the classical two parameter Bailey Lemma is proved, and a basic one parameter BCn Bailey Lemma is obtained as a limiting case. Several summation and transformation formulas associated with the root system BCn 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 BCn 6 phi(5) summation formula, a similar program is followed to prove an infinite family of D-n Euler Pentagonal Number Theorems.