组合构造法在Rogers-Ramanujan型等式中的应用

Rogers-Ramanujan型等式是整数分拆理论与q-级数中最重要的研究内容之一。Rogers-Ramanujan型等式具有很多推广和限制形式，很多经典的等式都是Rogers-Ramanujan型等式，如Euler等式和Schur等式等。Rogers-Ramanujan型等式在数论、统计学、代数学等领域有着十分广泛的应用。因此，发现和证明新的Rogers-Ramanujan型等式始终是一个非常活跃的课题并且具有十分重要的价值和意义。本项目拟主要利用组合构造法，并结合贝利变换和q-差分方程研究Rogers-Ramanujan型等式，力争在三个重要问题上取得突破：Bressoud猜想、Rogers-Ramanujan型等式的奇偶性因素、以及Capparelli等式和Kanade-Russell猜想。

The Rogers-Ramanujan type identities are one of the most important research contents in the theory of partitions and q-series. There are many generalizations and refinements of the Rogers-Ramanujan type identities, and many classical identities are the Rogers-Ramanujan type identities, including Euler's identity, Schur's identity and so on. The Rogers-Ramanujan type identities are widely applied into number theory, statistics, algebra, and several other areas. Therefore, discovering and proving new Rogers-Ramanujan type identities is always a very active research topic and is of great value and significance. In this research project, the applicant aims to make breakthroughs in Bressoud's conjecture, parity considerations in the Rogers-Ramanujan type identities, and Capparelli's identities and Kanade-Russell conjectures by mainly using combinatorial constructions, and combined with the Bailey transform and q-difference equations.