Overpartition analogues of q-binomial coefficients and their applications
In partition theory it is well known that the Gaussian polynomials ( or q binomial coefficients) [m:,n] generate integer partitions that fit inside an mxn rectangle, that is, partitions with largest part at most m and the number of parts does not exceed n. In this dissertation , we study the overpartition extension of these polynomials known as over q-binomial coefficients. These are ge11 erati11g fuuctio11s that enumerate overpartitio11s fitting inside au m, x n rectangle. First we discuss the ordinary q-binomial coefficients. Then we examine the over q-hiuomial coefficients with respect to their motivations, q-analogues of classical identities, and various combinatorial and analytic applications.
A dissertation submitted in fulfillment of the requirement of the degree of Master of Science (M.Sc.) in Pure Mathematics to the Faculty of Science, University of the Witwatersrand, 2018