Identities on Zagier's rank two examples for Nahm's problem

2022/10/19 by Wang, Liuquan · 5 citations
#11F03 #11P84 #33D15 #33D60 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2210.10748

Abstract

Let r≥ 1 be a positive integer, A a real positive definite symmetric r× r matrix, B a vector of length r, and C a scalar. Nahm's problem is to describe all such A,B and C with rational entries for which a specific r-fold q-hypergeometric series (denoted by f_A,B,C(q)) involving the parameters A,B,C is modular. When the rank r=2, Zagier provided eleven sets of examples of (A,B,C) for which f_A,B,C(q) is likely to be modular. We present a number of Rogers--Ramanujan type identities involving double sums, which give modular representations for Zagier's rank two examples. Together with several known cases in the literature, we verified ten of Zagier's examples and give conjectural identities for the remaining example.

Cited by

• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Tadpole Nahm sum as a Wronskian
• Nahm sum identities for Cartan matrices of type D_k
• On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples

Related

• Explicit Forms and Proofs of Zagier's Rank Three Examples for Nahm's Problem
• Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums
• Counterexamples to Zagier's Duality Conjecture on Nahm Sums
• Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples
• Some New Modular Rank Four Nahm Sums as Lift-dual of Rank Three Examples