贝利变换及其在整数分拆上的应用

Q-series includes many aspects, such as theta functions, partial theta functions and mock theta functions, which is closely related to the theory of partitions. At the Millennial Conference on Number Theory, George Andrews challenged mathematicians in the 21st century to elucidate the overlap between classes of q-series and modular forms, which has its origin in mock theta functions. The Bailey transform plays a crucial role in q-series, especially in Rogers-Ramanujan type identities. Recent studies indicate that the Bailey transform can also apply to the study of mock theta functions. Therefore, the Bailey transform and mock theta functions have received great attentions from experts at home and abroad. In this project, with the aid of the Bailey transform, the applicant aims to study mock theta functions and Rogers-Ramanujan type identities, and further relate these to partition identities and spt type functions. It is hoped that the implementation of the project will deepen the understanding of mock theta functions, Rogers-Ramanujan type identities, and their connection with partitions, thus will further promote the joint development of the theory of partitions and q-series.

q-级数包括西塔函数，部分西塔函数和仿西塔函数等许多方面，它与整数分拆理论有着十分密切的联系。在2000年千禧年数论大会上，G.E. Andrews就q-级数的分类和模形式之间的交叉问题向数学家们提出了挑战，而这一挑战的源头正是仿西塔函数。贝利变换是q-级数，尤其是Rogers-Ramanujan型等式的重要研究工具。近期的研究表明，贝利变换对于仿西塔函数的研究也有着十分重要的作用。因此，贝利变换和仿西塔函数受到了国内外专家的高度关注和重视。本项目拟利用贝利变换来研究仿西塔函数和Rogers-Ramanujan型等式，并将它们与分拆恒等式和spt型分拆函数建立联系。项目的实施将有助于加深对仿西塔函数和Rogers-Ramanujan型等式的了解，以及它们与整数分拆的联系，进而促进整数分拆理论和q-级数的共同发展。

q-级数与整数分拆理论有着十分密切的联系。贝利变换又在q-级数的研究中发挥着十分重要的作用。本项目围绕贝利变换以及它在整数分拆中的应用展开研究，并在以下几个方面取得了重要进展。首先，对于美国数学协会前主席D.M. Bressoud在1980年提出的猜想，我们给出了它的overpartition模拟，并最终解决了该猜想中j=0的情形。其次，我们推广了overpartition中D-秩的定义，该推广定义也将普通分拆中的c-秩，秩，k-秩统一了起来。此外，通过研究q-级数展开式中系数的性质我们还提供了一种可以用于证明q-级数恒等式和同余式的统一方法，并且该方法还可以用于发现新的结果。以上相关结果分别发表在Advances in Mathematics, International Journal of Number Theory和Experimental Mathematics上。