An elliptic 𝐵𝐶_{𝑛} Bailey Lemma, multiple Rogers–Ramanujan identities and Euler’s Pentagonal Number Theorems

Coskun, Hasan

doi:10.1090/S0002-9947-08-04457-7

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.

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