关于截断超几何函数及其相关序列的超同余式

我们将研究关于截断超几何函数及其相关序列的超同余式。近年来，此课题已成为组合数论方面的一个活跃研究领域。对其的研究需要综合运用组合数学、数论、分析、代数等数学分支中的工具。我们将聚焦于三个方面的问题：.⑴ 用局部-整体方法得到超几何恒等式的p-adic模拟。局部-整体方法是我们最近用于处理截断超几何函数的新工具，其优势在于能避免对导数的讨论。我们还将研究此方法对q-同余式的应用。.⑵ 从同余式出发发现新的级数恒等式。受同余式与无穷级数之间的对应关系启发，我们计划寻找一些无穷级数恒等式的参数型推广。我们还将从同余式出发研究涉及Gauss超几何函数的恒等式。.⑶ 研究由Calabi-Yau方程生成序列的Dwork型同余式。Dwork型同余式起源于Dwork对正则幂级数p-adic连续性的工作。Calabi-Yau方程是一类特殊常微分方程。此类方程特解所对应的序列常满足一定的Dwork型同余式。

We plan to research the supercongruences concerning the truncated hypergeometric functions and the related sequences. In recent years, this topic has become one of active research areas in combinatorial number theory. In order to study the truncated hypergeometric functions, we have to use the tools from combinatorics, number theory, analysis, algebra etc.. We shall focus on the following three problems...(1) Use the local-global method to establish the p-adic analogues of hypergeometric identities. The local-global method is a new technique to deal with the truncated hypergeometric functions. The advantage of this method is that the discussions on the derivatives can be avoided. Furthermore, we shall consider the applications of the local-global method to q-congruences...(2) Find new identities concerning infinite series from the viewpoint of congruences. Motivated by the corresponding relation between congruences and infinite series, we shall try to search the generalizations with parameters of some classical identities concerning infinite series. Moreover, we shall research the identities on Gaussian hypergeometric functions, from the viewpoint of congruences...(3) Study Dwork-type congruences concerning the sequences arising from Calabi-Yau equations. The Dwork-type congruences origins from Dwork’s work on the p-adic continuity of formal power series. The Calabi-Yau equation is a class of ordinary differential equations. The sequences corresponding to the solutions of Calabi-Yau equations often satisfy some Dwork-type congruences.