Ghost series and a motivated proof of the Bressoud–Göllnitz–Gordon identities

We present what we call a “motivated proof” of the Bressoud-Göllnitz-Gordon identities. Similar “motivated proofs” have been given by Andrews and Baxter for the Rogers–Ramanujan identities and by Lepowsky and Zhu for Gordon’s identities. Additionally, “motivated proofs” have also been given for the Andrews-Bressoud identities by Kanade, Lepowsky, Russell, and Sills and for the Göllnitz–Gordon–Andrews identities by Coulson, Kanade, Lepowsky, McRae, Qi, Russell, and the third author. Our proof borrows both the use of “ghost series” from the “motivated proof” of the Andrews–Bressoud identities and uses recursions similar to those found in the “motivated proof” of the Göllnitz–Gordon–Andrews identities. We anticipate that this “motivated proof” of the Bressoud–Göllnitz–Gordon identities will illuminate certain twisted vertex-algebraic constructions.