The Applications Of Bailey’s Lemma To The Rogers-Ramanujan Type Identities

In the development history of q-series for more than 200 years,RogersRamanujan type identities have always been an important research topic of q-series.Mac Mahon provided the combinatorial interpretation of RogersRamanujan identities using the method of combinatorial construction.Such partition theorem includes the famous Euler’s partition theorem,Schur’s theorem and the G (?)llnitz-Gordon theorem,whose corresponding algebraic forms were also called the Rogers-Ramanujan type identities.In 1980,Bressoud obtained an algebraic extension of the Rogers-Ramanujan identities,which includes many classical Rogers-Ramanujan type partition identities,such as Andrews-Gordon identity,Bressoud-Rogers-Ramanujan identity,AndrewsG (?)llnitz-Gordon identity,and Bressoud-G (?)llnitz-Gordon identity,and so on in addition to all of above identities.Bailey pairs and Bailey’s lemma are important tools of the theory of q-series.Bailey and Slater using them produced many Rogers-Ramanujan type identities.This article mainly reviews the applications of Bailey’s lemma to the Rogers-Ramanujan type identities,which specially gives a new algebraic proof of Bressoud’s generalizations of the Rogers-Ramanujan identities by using Bailey’s lemma.In chapter 1,we mainly introduce some basic concepts,theorems and some classical Rogers-Ramanujan type identities in the theory of integer partition and q-series.Chapter 2 is devoted to reviewing the proofs of some classical RogersRamanujan type identities,including the Rogers-Ramanujan identities,the Rogers-Ramanujan-Gordon theorem,the Bressoud-Rogers-Ramanujan-Gordon theorem and the Andrews-G (?)llnitz-Gordon theorem.In the third chapter,we mainly introduce Bailey pairs,Bailey’s lemma and related knowledge,and the applications they exert in proving the Rogers-Ramanujan identities,Andrews-Gordon identity,Bressoud’s identity and Andrews-G (?)llnitz-Gordon identity.Using Bailey pairs,Bailey’s lemma and change of base formulas,we complete the algebraic proof of the Bressoud’s generalizations of the RogersRamanujan identities in Chapter 4.