Andrews-G(?)llnitz-Gordon Type Theorems For Overpartitions

The integer partition theory is a very important part of the q-series theory,by the theory of partition we can give combinatorial proofs of many difficult basic hypergeometric identities intuitively.In the theory of partitions,many theorems are concerned about the equality between the numbers of two types of partitions,namely for any nonnegative integer n,the numbers of partitions of n are equal under the two types of restrictions.The most famous theorem of this type is the Rogers-Ramanujan theorem.Gordon,Andrews and Bressoud gave the combina-torial and analytic generalization of the Rogers-Ramanujan theorem.Chen,Sang and Shi also obtained the overpartition analogues of the generalized theorems.The G(?)llnitz-Gordon theorem is another important theorem,Andrews obtained a combinatorial generalization of the G(?)llnitz-Gordon theorem,called the Andrews-G(?)llnitz-Gordon theorem.After that,Bressoud gave an analytic generalization of the G(?)llnitz-Gordon identities,which can be seen as the generating function form of the Andrews-G(?)llnitz-Gordon theorem.The main results in this paper are the combinatorial and analytic overpartition analogues of the Andrews-G(?)llnitz-Gordon theorem.The full text is divided into six parts:In the first chapter,we give a brief introduction of the research background and status as well as the main results and structure of this paper.In the second chapter,we recall some basic concepts and symbols,then intro-duce the Rogers-Ramanujan identities,the Rogers-Ramanujan-Gordon theorem and the overpartition analogue of the Rogers-Ramanujan-Gordon theorem.In the third chapter,we first introduce the G(?)llnitz-Gordon identities and the Andrews-G(?)llnitz-Gordon theorem,then give the main results of this paper,namely the combinatorial and analytic overpartition analogues of the Andrews-G(?)llnitz-Gordon theorem.By using Bailey pair,Bailey's lemma and the change of base formula,we give a proof of the analytic overpartition analogue of the Andrews-G(?)llnitz-Gordon theorem in the fourth chapter.In the fifth chapter,we first recall the notion of the Gordon marking,then introduce the definition of the G(?)llnitz-Gordon marking of an overpartition,and define the clusters based on it.Next,we give two pairs of inverse operations on overpartitions,based on the operations we construct two bijections to complete the proof of the combinatorial overpartition analogue of the Andrews-G(?)llnitz-Gordon theorem.The last chapter is devoted to summarizing the whole paper and show the future research direction.