Mathematical Sciences: Hypergeometric Functions over Finite Fields

本文研究有限域上超几何级数的特征标和类比。主要研究人员将系统地研究贝利变换的一个类比，试图推导出更高阶超几何级数恒等式的类比。他还将研究“伪”超几何级数，即由于有限域的性质，函数不能写成特征和超几何级数，而是密切相关的。主要研究人员还将研究超几何级数与表示论之间的关系。前两个问题的目的是完善有限域上超几何级数的理论，提高其作为特征和计算工具的计算能力。最后一个问题的目的是发展这两个数学领域之间的相互作用，将表征理论工具应用于特征和问题，并确定对已知的特征和恒等式的表征理论解释。

This research concerns character sum analogues for hypergeometric series over finite fields. The principal investigator will systematically study an analogue of the Bailey transform in an attempt to derive analogues for higher order hpergeometric series identities. He will also study "pseudo" hypergeometric series, that is, functions which due to the nature of the finite field can not be written as a character sum hypergeometric series but are closely related. Also the principal investigator will investigate the relationship between hypergeometric series and representation theory. The purpose of the first two problems is to complete the theory of hypergeometric series over finite fields and to increase its computational power as a tool for character sum evaluation. The goal of the last problem is to develop an interplay between the two areas of mathematics, to bring representation theoretic tools to bear on character sum problems and to determine representation theoretic interpretations for known character sum identities.