turkmath.org

One branch of integer partition theory studies partition identities, such as Euler’s partition identity, Rogers-Ramanujan identities, and Capparelli’s identities to name some milestones. The majority of identities relate gap conditions to residue conditions. Gap conditions control how many times a part can appear in the partitions, and residue conditions stipulate which parts can appear without a bound on the number of repetitions. Kanade and Russell in 2015 developed a computerized method to discover partition identities, and made a fundamental change in our paradigm. In a PURE project in the summer of 2021, students implemented a variant of Kanade and Russell’s IdentityFinder to discover partition identities with a novel theme, namely ”eventually periodic residue conditions”. Most of these identities can be proven by elementary means, some of them require transformations to be recognized as corollaries of known results. There is one new construction of an evidently positive generating function due to a lucky mistake. This is the preliminary report on joint work with Salih Numan B¨uy¨ukba¸s, Vahit Alp Hıdıro˘glu, and Omer Surhay Kocakaya, ¨ our undergraduate students