Overpartition Analogues Of Rogers-Ramanujan Type Identities For Even Moduli

Rogers-Ramanujan identities is one of the most charming part in the theory of integer partitions.Gordon obtained a combinatorial generalization of the Rogers-Ramanujan identities,so called the Rogers-RamanujanGordon identities.Bressoud gave an analytic generalization of the RogersRamanujan-Gordon identities,which specializes to many classical RogersRamanujan type identities,including the G?llnitz-Gordon identities for even moduli which can be called the Bressoud-G?llnitz-Gordon identites.Recently,many researches investigated the overpartition analogue of classical Rogers-Ramanujan type identities.In this thesis,we will first give an overpartition analogue of Bressoud-G?llnitz-Gordon identites.Then,we will give an overpartition of Bressoud's generalization of the Rogers-RamanujanGordon identities for even moduli,which can specialize to another overpartition analogue of Bressoud-G?llnitz-Gordon identites.In Chapter 1,we introduce the notation of integer partitions.We then recall some classical Rogers-Ramanujan type identities,such as the RogersRamanujan-Gordon identities,Bressoud-G?llnitz-Gordon identities and Bressoud's generalization of the Rogers-Ramanujan-Gordon identities.Their overpartition analogue will also presented.By using Bailey's lemma and change of base formulas due to Bressoud,Ismail and Stanton,we give the analytic parts of the overpartition analogues of Bressoud-G?llnitz-Gordon identities and Bressoud's generalization of the Rogers-Ramanujan-Gordon identities for even moduli in Chapter 2.In Chapter 3,we begin with the notion of the G?llnitz-Gordon marking of an overpartition and then give an outline of the proof of the combinatorial part of the overpartition analogue analogue of Bressoud-G?llnitz-Gordon identities,which reduces to two lemmas stated at the end of this chapter.Chapter 4 is devoted to the bijective proofs of the two lemmas stated at the end of chapter 3,and then complete the proof of combinatorial part of the overpartition analogue analogues of Bressoud-G?llnitz-Gordon identities.In Chapter 5,we present a bijective proof of the combinatorial part of the overpartition analogue of Bressoud's generalization of the RogersRamanujan-Gordon identities for even moduli.