Combinatorial Interpretation of Rogers Ramanujan Identities and Their Analogues

Abstract

In this thesis, we study about partitions of positive integers. The study of parti- tions of positive integers has fascinated a number of great mathematicians: Euler, Legendre, Ramanujan, Hardy, Rademacher, Sylvester and Dyson. Here, we start from the basic q-series and then move quickly to one of the most magni cent and surprising results of the entire subject, the Rogers-Ramanujan identities. In Chapter 1, we give a brief historical background of the subject matter and give some de nitions, which we shall be using in this thesis. In Chapter 2, we give analytic proof of Rogers-Ramanujan identities using Bailey's Lemma and a combinatorial proof of Rogers-Ramanujan identities due of MacMo- han. Finally in Chapter 3, we prove some more q-series identities combinatorially, ana- logues to Rogers-Ramanujan identities using ordinary partition function.