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The theory of integer partitions is a field of much investigative interest to mathematicians and physicists. Unlike some of the more well-known sequences such as the Fibonacci Numbers, integer partitions follow no obvious discernible pattern. Mathematicians have been left to identify certain recurring patterns within the series: certain identities relating partitions satisfying seemingly unrelated conditions. Arguably, the most famous of these identities are those attributed to mathematicians Srinivasa Ramanujan and Leonard James Rogers. The Rogers-Ramanujan Identities have, in many ways, laid the ground work for much of the research being conducted in this field in recent history. In May of 1989, George Andrews and Rodney J. Baxter published what they called a “motivated proof” of the Rogers-Ramanujan identities. It is in the spirit of this motivated proof that we conduct our research.
We look to answer the following question: Can we complete a proof of an “Empirical Hypothesis” in the spirit of the Andrews-Baxter motivated proof in order to discover identities of the Rogers-Ramanujan type in regard to overpartitions? Through the course of our research, we will prove an “Empirical Hypothesis” and subsequently identify new partition and overpartition identities. What distinguishes our research from the research done previously is the application to overpartitions. Overpartitions have appeared in recent work of many mathematicians, and their study bears strong parallels to the study of partitions. We hope to extend the work on “motivated proofs” of partition identities to overpartitions, and to find new overpartition identities using this technique.
Takita, Collin R., "Results on Ghost Series and Motivated Proofs of Overpartition Identities" (2015). Mathematics Summer Fellows. 1.