Construction of evidently positive series and an alternative construction for a family of partition generating functions deu to Kanade and Russell

Construction of generating functions for partitions, especially construction evidently positive series as generating functions for partitions is a quite interesting problem. Recently, Kurşungöz has been constructed evidently positive series as generating functions for the partitions satisfying the di erence conditions imposed by Capparelli's identities and Göllnitz-Gordon identities, for the partitions satisfying certain di erence conditions in six conjectures by Kanade and Russell and the partitions satisfying the multiplicity condition in Schur's partition theorem. In this thesis, we give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions which are evidently positive series for partitions due to Kanade and Russell. To obtain those generating functions, we rst construct an evidently positive series for a key in nite product. In that construction, a series of combinatorial moves is used to decompose an arbitrary partition into a base partition together with some auxiliary partitions that bijectively record the moves