On a Partition Theorem of Göllnitz and Quartic Transformations

Abstract

Fori=1, 2, 3, 4, letQi(n) denote the number of partitions of n into distinct parts ≢ (mod4). New weighted identities in three free parameters are established connectingQi(n) with partitions whose parts differ by ⩾4 and such that consecutive members of the arithmetic progression ≡i(mod4) cannot occur as parts. By the use of suitable quartic transformations, these weighted identities are shown to be reformulations of a deep partition theorem of Göllnitz. Applications include new relations for partitions of the Göllnitz-Gordon type, a new proof of Jacobi's triple product identity and a remarkable congruence modulo powers of 2 forQ2(n).