A Survey On Riordan Group

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.