Göllnitz-Gordon函数及其在数论中的应用

The theory of G?llnitz-Gordon functions is one of the most active fields in contemporary combinatorics and q-series, which has numerous natural connections to other branches of mathematics, that is exactly the reason why many well-known combinatorialists, including Professor George Andrews who is the president of the American Mathematical Society, are interested in it.. This project focuses on establishing new identities involving G?llnitz-Gordon functions and giving new proofs of the currently known identities by mathematics mechanization and the theory of Ramanujan''s modular equations. We shall apply the recurrence relations from mathematics mechanization to construct the combinatorial proofs of these identities. Using the theories of mathematics mechanization and theta functions, we shall investigate the applications of G?llnitz-Gordon functions in number theory and give number-theoretic interpretations of these results. . In the process of executing this project, we anticipate that by the methods mentioned in this project, we can not only discover some new identities, but also eventually intensify the bonds to mathematics mechanization.

G?llnitz-Gordon函数理论是当前组合数学和q-级数的研究热点之一。该课题与其它多个数学分支有着重要的联系，吸引了包括美国数学会会长George Andrews教授在内的众多知名组合学家的研究兴趣。. 本项目旨在利用数学机械化手段和Ramanujan模方程理论建立新的包含G?llnitz-Gordon函数的恒等式或给出已知恒等式新的证明方法；将数学机械化方法的递归思想应用于组合证明，构造相关恒等式的组合证明；综合运用数学机械化方法和theta函数理论，研究G?llnitz-Gordon函数在数论中的应用并给出相关结果的数论解释。. 通过项目的实施，我们希望利用申请书中的方法不仅能够发现更多新的恒等式，而且还可以进一步深化该领域与数学机械化的联系。

Göllnitz-Gordon函数是当前q级数的研究热点之一，该函数与其它多个数学分支有着重要的联系。本项目主要研究对象是Göllnitz-Gordon函数、theta函数及其在数论中的应用。我们发表或接受发表了6篇论文，其中3篇发表，3篇接受发表。我们的文章发表或接受发表于J. Number Theory，Inter. J. Number Theory, Ramanujan. J, Ars Combin. 和Bull. Austr. Math. Soc。我们利用Göllnitz-Gordon理论和theta函数分块公式发现了多个关于overpartitions的模9和模27的同余关系，推广了Hirschhorn和Sellers的结果。利用数学机械化方法和theta函数理论，我们确定了一类eta商函数的Fourier展开式的系数，推广了Williams的结论。利用生成函数和theta函数分块公式，我们研究了9-regular partitions的算术性质，得到了9-regular partitions的奇偶性。我们解决了由Gasarch, Moriarty 和Tumma提出的一个关于“2-color Rado Numbers”猜想。利用mock theta 函数理论，我们建立了多个关于1-shell Totally Symmetric Plane Partitions的模4和模8的同余性质，回答了Hirschhorn和Sellers提出的一个公开问题。