The Rogers-Ramanujan Identities Explored
From Ramanujanā€™s Proof to Modern Bijective Approaches
A.A.S. Autar (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Wolter Groenevelt ā€“ Mentor (TU Delft - Analysis)
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Abstract
The Rogers-Ramanujan identities are among the most remarkable results in the theory of integer partitions and š¯‘˛-series. Discovered independently by Rogers, Ramanujan, and Schur, these identities have since appeared in a wide range of mathematical disciplines, including number theory, combinatorics, representation theory, and statistical mechanics. Despite their concise form, the identities have inspired dozens of proofs and interpretations, each revealing new layers of structure and insight.
This thesis investigates four core aspects of the Rogers-Ramanujan identities. First, it presents and explains Ramanujanā€™s original analytic proof. Second, it explores the combinatorial interpretation of the identities in terms of integer partitions. Third, the thesis provides a chronological overview of key combinatorial bijective proofs, highlighting how these have evolved and deepened our understanding of the identities. Lastly, it presents Igor Pakā€™s combinatorial proof of the first Rogers-Ramanujan identity.
By combining historical context with mathematical exposition, this work aims to clarify the richness of the Rogers-Ramanujan identities and to contribute to the ongoing effort to understand them more fully.