New developments in hypergeometric equations

超几何方程的新进展

• 批准号: 1001763
• 负责人: Laura Matusevich
• 金额: $ 15万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001763&HistoricalAwards=false
• 关键词: New; developments; hypergeometric; equations

Hypergeometric functions are power series (in one or several variables) whose coefficients satisfy a two-term recursion. Their very definition makes hypergeometric functions amenable to treatment by analytic and combinatorial methods. The study of their differential equations from a D-module theoretic point of view is the subject of this project. A breakthrough in the late 1980s due to Gelfand, Graev, Kapranov and Zelevinsky provided a new connection between hypergeometric functions (in several variables) and the theory of toric varieties, a subfield of algebraic geometry with a strong combinatorial flavor. This link has made available a variety of algebraic, homological and combinatorial tools to study hypergeometric differential equations and their solutions. The goal of this project is to advance the theory of hypergeometric functions and differential equations.The solutions of hypergeometric differential equations comprise a rich and interesting class of functions. Familiar functions, such as the trigonometric functions, belong to this class. The importance of these functions lies in their usefulness, not only within mathematics, but in physics and engineering as well. The present research will provide advances in the theory of hypergeometric differential equations; its underlying philosophy is to view hypergeometric functions as bridges between different areas of mathematics.

超几何函数是幂级数（一个或多个变量），其系数满足两项递归。它们的定义使得超几何函数可以用分析和组合方法来处理。从D-模理论的角度研究它们的微分方程是这个项目的主题。20世纪80年代后期，由于Gelfand，Graev，Kapranov和Zelevinsky的突破，提供了超几何函数（多变量）和复曲面簇理论之间的新联系，这是代数几何的一个子领域，具有强烈的组合风味。这个链接提供了各种代数，同调和组合工具来研究超几何微分方程及其解决方案。本项目的目标是推进超几何函数和微分方程的理论，超几何微分方程的解包含了丰富而有趣的函数类。熟悉的函数，如三角函数，属于这一类。这些函数的重要性在于它们的实用性，不仅在数学中，而且在物理学和工程学中也是如此。本研究将提供超几何微分方程理论的进展，其基本哲学是将超几何函数视为不同数学领域之间的桥梁。