Analytic and Combinatorial Aspects of Certain Identities and Congruences in Partition Theory

Download or read book Analytic and Combinatorial Aspects of Certain Identities and Congruences in Partition Theory written by Shreejit Bandyopadhyay and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation explores three related topics in the theory of integer partitions. We first revisit certain conjectures made by George Beck about the crank of a partition. We discuss a new approach to these conjectures by decomposing the relevant crank generating function and further explore connections with certain identities involving generalized Lambert series and Appell-Lerch sums. After that, we derive certain weighted generating functions of cranks by both analytic and combinatorial arguments. Lastly, we consider overpartition analogues for Bressoud's generalizations of the famous Rogers-Ramanujan identities. We discuss some new Rogers-Ramanujan type overpartition identities under certain constraints on odd and even parts of the partitions.