The Dual of Göllnitz’s (big) Partition Theorem*

A Rogers-Ramanujan (R-R) type identity is a q-hypergeometric identity in the form of an infinite (possibly multiple) series equals an infinite product. The series is the generating function of partitions whose parts satisfy certain difference conditions, whereas the product is the generating function of partitions whose parts usually satisfy certain congruence conditions. For a discussion of a variety of R-R type identities, see Andrews [16], Ch.9. The partition theorem which is the combinatorial interpretation of a q-hypergeometric identity, is called a Rogers-Ramanujan type partition identity. A q-hypergeometric R-R type identity is usually discovered first and then its combinatorial interpretation as a partition theorem is given. There are important instances of Rogers-Ramanujan type partition identities being discovered first and their q-hypergeometric versions given later. Perhaps the first such significant example is the 1926 partition theorem of Schur [24]: Theorem S: Let T (n) denote the number of partitions of an integer n into parts ≡ ±1 (mod 6). Let S(n) denote the number of partitions of n into distinct parts ≡ ±1 (mod 3). Let S1(n) denote the number of partitions of n into parts that differ by ≥ 3, where the inequality is strict if a part is a multiple of 3. Then