【作者】 任利梅；

【导师】 马欣荣；

【作者基本信息】 苏州大学 ， 应用数学， 2009， 硕士

【摘要】 本文主要就Lagrange反演公式、Fa(?) di Bruno公式和Riordan群各自理论形成、内容方法以及彼此之间的联系和区别所做的一个综述。第一章总结了Riordan群的各种主要定义以及与它相关的Lagrang反演公式和Fa(?) di Bruno公式。第二章对Riordan群理论的主要运算规则、目前已经建立的Riordan群的主要理论做一概述,其中由Lagrange反演公式给出的运算法则对级数求和有着重要作用。第三章主要分析了Riordan群在计算组合和式方面以及反演关系方面的应用,并且纠正了Egorychev等人文章中的一个组合反演关系,推广了H.Wilf的著作《Generating Functionology》中一个问题结论。第四章通过详细计算阐述了这样一个历史误会(这也是本篇综述的目的所在):Riordan群不是人们理解成的一个新方法,其思想早就蕴含在一个多世纪前就存在的Fa(?) di Bruno公式中了。最后一章提出了一个尚待解决的问题。更多还原

【Abstract】 In this thesis,we try to make a comprehensive summary on Riordan group as well as its interplay with the celebrated Lagrange inversion formula,the famous Fa(?) di Bruno formula and their various applications in Combinatorial Analysis.Chapter one we summarize all main concepts of the Riordan group and Riordan array and its connections with the Lagrange inversion formula Fa(?) di Bruno formula.Chapter two is devoted to all main results in the theory of the Riordan group obtained up to date,some frequently used techniques invoking the Lagrange inversion formula are also rephrased.In the third chapter,we skctch three aspects of applications of the Riordan group, more precisely,in combinatorial summations and their dual forms in viewpoint of inverse relations.In addition,we give the correct form of an identity poised by Egorychev et al.with an evident error,and a generalization of an identity recorded by Wilf in his book "Generatingfunctionology".In Chapter four,as a basic motivation of this thesis,we put forward a novel opinion on a historical misunderstanding by a direct calculation:the Riordan group is just a special case of the Fa(?) di Bruno formula,which can date back to more than one and half century ago.The final chapter we pose a new kind of recursive relations as an open problem.更多还原