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

TL;DR

This paper investigates specific modular $q$-hypergeometric series associated with Zagier's rank two examples of Nahm's problem, providing new identities and verifying most of Zagier's cases.

Contribution

The paper presents Rogers--Ramanujan type identities for Zagier's rank two examples, verifying ten cases and proposing conjectural identities for the remaining one.

Findings

Verified ten of Zagier's rank two examples as modular series.

Derived new Rogers--Ramanujan type identities involving double sums.

Proposed conjectural identities for the remaining Zagier example.

Abstract

Let $r\geq 1$ be a positive integer, $A$ a real positive definite symmetric $r\times r$ 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.