An Elliptic BC_n Bailey Lemma, Multiple Rogers--Ramanujan Identities and Euler's Pentagonal Number Theorems

2006/05/24 by Hasan Coskun, Coskun, Hasan
Mathematics · #05A19 #05E20 #11B65 #33D67 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A19 #msc:05E20 #msc:11B65 #msc:33D67

paper · pdf · doi:10.48550/arxiv.math/0605653

V2: 36 pages; to appear in AMS Trans; references added; typos corrected

arxiv created 2006/11/22 · arxiv updated 2009/12/01

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 ₆ϕ₅ 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}ϕ₅ summation formula, a similar program is followed to prove an infinite family of D_n Euler's Pentagonal Number Theorems.

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• Two new Bailey pairs and their q-identities of Rogers-Ramanujan type modulo 15, 24, and 30
• Shifted versions of the Bailey and well-poised Bailey lemmas
• Rogers-Ramanujan type identities and Chebyshev Polynomials of the third kind
• Bilateral Bailey pairs and Rogers-Ramanujan type identities