An A2 Bailey Lemma and Rogers–ramanujan-type Identities

for |q| < 1. The fame of these identities lies not only in their beauty and fascinating history [17, 3], but also in their relevance to the theory of partitions and many other branches of mathematics and physics. In particular, MacMahon [27] and Schur [39] independently noted that the left-hand side of (1.1) is the generating function for partitions into parts with difference at least two, while the right-hand side generates partitions into parts congruent to ±1 modulo 5. Similarly, the left-hand side of (1.2) is the generating function for partitions into parts with difference at least two and no parts equal to 1, while the right-hand side of (1.2) generates partitions into parts congruent to ±2 modulo 5. Over the years many generalizations of both the analytic and the combinatorial statement of the Rogers–Ramanujan identities have been found (see, e.g., [16, 2, 7, 8, 9, 4]). All the cited analytic generalizations are accessible through the classical, or A1, Bailey lemma and can thus be classified as “A1 Rogers–Ramanujan-type identities”. (We always mean identities of the “sum=product” form when referring to Rogers–Ramanujan-type identities.) The proof of Bailey’s lemma relies on a 3φ2 summation known as the q-Pfaff– Saalschütz formula. In [30, 31] Milne and Lilly used a 6φ5 sum for An−1 basic hypergeometric functions to establish a higher-rank version of Bailey’s lemma. Though