Construction of series as generating functions and verification type proofs for rogers-ramanujan generalization for partitions and overpartitions

Official URL: https://risc01.sabanciuniv.edu/record=b2767346_(Table of contents)

During the last century, researchers studied integer partition theory extensively. We are more interested in exploring partition identities among many aspects of integer partitions. In this thesis, we study a constructive method developed by Kur sungoz to nd new identities on Rogers-Ramanujan type integer partitions and overpartitions. For this aim, we give a reproof of two Rogers-Ramanujan identities using the constructive method. Combining two types of partitions, we introduced 2-colored Rogers-Ramanujan partitions. By nding some functional equations and using the constructive method, some identities have been found. Our results coincide with some extreme cases of Rogers-Ramanujan-Gordon's identities. A correspondence between colored partitions and those overpartitions is provided. Our second result is nding the missing cases of parity consideration on Rogers- Ramanujan-Gordon's identities due to Andrews's suggestion in his seminal paper about parity in partition identities. Four cases had proven by Sang, Shi, and Yee, we reproved them using the said constructive method and then found and proved the remaining cases by the same method.