超幾何算子之 ZETA 行列式

Abstract

本文研究高斯超幾何方程的施圖姆-劉維爾形式及其Zeta 行列式.基於Lesch 給出以wronskian 表示正則奇異施圖姆-劉維爾算子之Zeta 行列式的公式,以及拉馬努金的某項公式,便得出本文的主要結果: 超幾何方程之施圖姆-劉維爾形式的Zeta 行列式之closed form.

In the thesis, the eignevalue probelm of the Hypergeometric equation(for short: HGE or E(a,b,c)) is discussed. There are three parts in this thesis.First of all, I introduce some heuristic backgrounds and motivations of the eigenvalue problem of HGE. The second part is a survey about the theory of the HGE. Finally, based on Lesch’s formula of zeta determinant of Regular Singular Sturm-Liouville Operators, I calculate the zeta determinant with repect to the Sturm-Liouville form of HGE operator on a closed interval, by using Ramanujan's identities.