Explicit forms and proofs of Zagier's rank three examples for Nahm's problem.

Let r ≥ 1 be a positive integer, A a real positive definite symmetric r × r rational matrix, B a rational vector of length r , and C a rational scalar. Nahm's problem is to find all triples (A , B , C) such that the r -fold q -hypergeometric series f A , B , C (q) : = ∑ n = (n 1 , ... , n r) T ∈ (Z ≥ 0) r q 1 2 n T A n + n T B + C (q ; q) n 1 ⋯ (q ; q) n r becomes a modular form, and we call such (A , B , C) a modular triple. When the rank r = 3 , after extensive computer searches, Zagier provided twelve sets of conjectural modular triples and proved three of them. We prove a number of Rogers–Ramanujan type identities involving triple sums. These identities give modular form representations for and thereby verify all of Zagier's rank three examples. In particular, we prove a conjectural identity of Zagier. [ABSTRACT FROM AUTHOR]