On the Rogers-Ramanujan identities and partition congruences
Date
2024
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Abstract
In this dissertation, we study the Rogers-Ramanujan identities and partition congruences. The original Rogers-Ramanujan identities are proved analytically via Bailey’s construction. To the related Rogers-Ramanujan identities, our approach to the proof is via partition analysis and Bailey’s construction. The summation part of these Rogers-Ramanujan identities is established using partition analysis. Our work on congruences starts with a revisit of the popular Ramanujan’s congruences for the unrestricted partition function. The Atkin-Swinnerton-Dyer congruences for moduli 5 and 7 are obtained and the Ramanujan’s most beautiful identity is proved. The proof techniques for Ramanujan’s most beautiful identity are extended to another version in modulo 7. As part of the contribution to knowledge, a recurrence formula for the parity of the number of 2-color partitions of 2n in which one of the colors appears only in parts that are even is derived.
Description
A research report submitted in partial fulfilment of the requirements for the degree Master of Science to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2023
Keywords
Rogers-Ramanujan identities, Partition analysis