A complete generalization of Göllnitz’s “big” theorem

Abstract

In the spirit of Gollnitz’s “big” partition theorem of 1967, we present a new mod-6 partition identity. Alladi et al. provided a four-parameter refinement of Gollnitz’s big theorem in 1995 via a key identity of generating functions and the method of weighted words. By means of this technique, two similar mod-6 identities of this type were discovered—one by Alladi in 1999 and one by Alladi and Andrews in 2015. We finish the picture by presenting and proving the fourth and final possible mod-6 identity in this spirit. Furthermore, we provide a complete generalization of mod-n identities of this type. Finally, we apply a similar argument to generalize an identity of Alladi et al. from 2003.