STUDIES ON D-MODULES DERIVED FROM CONFLUENT HYPERGEOMETRIC DIFFERENTIAL EQUATIONS

合流超几何微分方程导出的D模研究

• 批准号: 09440052
• 负责人: MAJIMA Hideyuki
• 金额: $ 4.54万
• 依托单位: Ochanomizu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440052/
• 关键词: Confluent hypergeometric function; Irregukar singular; Irregularity; Asymptotic expansion; Period relation; Intersection theory; Divergent solution; Airy function; 合流超幾何方程式; D加群; 不確定特異点

1. Calculation of cohomology groups of solution complex of D-modules defined by confluent hypergeometric differential equations with values in the sheaves of germs of formal power-series ring and formal power-series ring with Gevrey order, by using projective resolutions of the D-modules similar to Koszul complex, which was invented by suggested information from 'KAN(a system' of computational algebraic analysis) made by Takayama2. Asymptotic expansions of restrictions of generalized Airy functions by using relations between confluent hypergeometric functions with particular parameters and generalized Airy functions3. Approximation formulas of coefficients of divergent solutions by using a vanishing theorem in asymptotic analysis in several variables4. Constructionf of theory of intersection on the complex projective line for homology and cohomology groups defined by connections which are regular singular or not, and quadratic relations satisfied by confluent hypergeometric functions, as an analogue of period relations, by applying this theory.

1. 通过使用 D-模的投影分辨率，计算由合流超几何微分方程定义的 D-模解复形的上同调群，其值为形式幂级数环和具有 Gevrey 阶的形式幂级数环的胚芽轮。类似于 Koszul 复合体的模块，它是根据 Takayama2 制作的“KAN（计算代数分析系统）”的建议信息发明的。利用具有特定参数的合流超几何函数与广义艾里函数之间的关系，渐近展开广义艾里函数的限制。多变量渐近分析中利用消失定理求发散解系数的逼近公式4．应用该理论，构建了由正则奇异或非正则奇异的连接定义的同调和上同调群的复射影线上的相交理论，以及合流超几何函数满足的二次关系，作为周期关系的类比。

项目成果

IWASAKI, Katsunori: "Asymptotic Analysis for Linear difference Equations" Transactions of the American Mathematical Society. 349・10. 4107-4142 (1997)

IWASAKI, Katsunori：“线性差分方程的渐近分析”美国数学会汇刊 349・10 4107-4142 (1997)。

"connecions" RIMS Kokyuroku. 1014. 31-35 (1997)

“连接”RIMS Kokyuroku。

伊勢 康代: "合流超幾何微分方程式のストークス現象について"お茶の水女子大学大学院人間文化研究科数理情報科学専攻(修士論文). 51 (1999)

伊势康世：《论合流超几何微分方程的斯托克斯现象》御茶水女子大学研究生院人文科学研究科数学信息科学系（硕士论文）51（1999）。

ISE, Yasuyo: "On Stokes Phenomenun of Confluent Hypergeometric Diffrential Equations"Master Thesis, Math. and Computer Sci., Graduate School of Humanities and Sciences[Master's Program], Ochanomizu University. 51 (1999)

ISE，Yasuyo：“论汇合超几何微分方程的斯托克斯现象”硕士论文，数学。

MAJIMA, Hdeyuki: "Some isomorphism theorems of cohomology groups for completely integrable connections"RIMS Kokyuroku. 1014. 31-35 (1997)

MAJIMA，Hdeyuki：“完全可积连接的上同调群的一些同构定理”RIMS Kokyuroku。