Ghost series and a motivated proof of the Andrews-Bressoud identities

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• 作者：Shashank Kanade^a ; ¹ ; ^{skanade@math.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; ^{kanade@ualberta.ca" class="auth_mail" title="E-mail the corresponding author} ; James Lepowsky^a ; ^{lepowsky@math.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; Matthew C. Russell^a ; ^{russell2@math.rutgers.edu" class="auth_mail" title="E-mail the corresponding author} ; Andrew V. Sills^b ; ^{asills@georgiasouthern.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Rogers&ndash ; Ramanujan-type identities ; Motivated proofs of partition identities ; Experimental mathematics
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：146
• 期：Complete
• 页码：33-62
• 全文大小：491 K

文摘

We present what we call a “motivated proof” of the Andrews–Bressoud partition identities for even moduli. A “motivated proof” of the Rogers–Ramanujan identities was given by G.E. Andrews and R.J. Baxter, and this proof was generalized to the odd-moduli case of Gordon's identities by J. Lepowsky and M. Zhu. Recently, a “motivated proof” of the somewhat analogous Göllnitz–Gordon–Andrews identities has been found. In the present work, we introduce “shelves” of formal series incorporating what we call “ghost series,” which allow us to pass from one shelf to the next via natural recursions, leading to our motivated proof. We anticipate that these new series will provide insight into the ongoing program of vertex-algebraic categorification of the various “motivated proofs.”