PARTITION CONGRUENCES AND DYSON’S RANK
DOI:
https://doi.org/10.29121/granthaalayah.v2.i2.2014.3066Keywords:
Modulo, Dyson’s Conjectures, Rank Of Partition, Ramanujan’s Lost Note Book, Theta SeriesAbstract [English]
In this article the rank of a partition of an integer is a certain integer associated with the partition. The term has first introduced by freeman Dyson in a paper published in Eureka in 1944. In 1944, F.S. Dyson discussed his conjectures related to the partitions empirically some Ramanujan’s famous partition congruences. In 1921, S. Ramanujan proved his famous partition congruences: The number of partitions of numbers 5n+4, 7n+5 and 11n +6 are divisible by 5, 7 and 11 respectively in another way. In 1944, Dyson defined the relations related to the rank of partitions. These are later proved by Atkin and Swinnerton-Dyer in 1954.
The proofs are analytic relying heavily on the properties of modular functions. This paper shows how to generate the generating functions for In this paper, we show how to prove the Dyson’s conjectures with rank of partitions.
Downloads
References
Andrews, G.E. and Garvan, F.G. ,Ramanujan’s Lost Notebook VI; the Mock Theta
Conjectures, Advances in Math., 73, (1989), 242–255. DOI: https://doi.org/10.1016/0001-8708(89)90070-4
Agarwal, A.K. ,Partitions Yesterday and Today, New Zealand Math. Soc., Wellington,(1979).
Andrews, G.E. (1979), An Introduction to Ramanujan’s Lost Notebook, Amer. Math. DOI: https://doi.org/10.2307/2321943
Monthly, 86: 89–108.
A.O.L.Atkin and H.P.F.Swinnerton-Dyer, Some properties of partitions,proc. London
Math.soc,(3),4 (1954).84-106.
F.J.Dyson, Some guesses in the theory of partitions, Eureka,8 (1944),10-15.
Garvan, F.G.,Generalizations of Dyson’s Rank, Ph. D. thesis, Pennsylvania State
University,(1986).
Garvan, F.G. (1988), Ramanujan Revisited, Proceeding of the Centenary Conference,
University of Illinois, Urban-Champion .
Garvan, F.G. (2013), Dyson’s Rank Function and Andrews’ spt-function,
University of Florida, Seminar Paper Presented in the University of Newcastle on
August (2013) .
S.Ramanujan, Congruences properties of partitions, Math. J.9 (1921), 147-153. DOI: https://doi.org/10.1007/BF01378341
Downloads
Published
How to Cite
Issue
Section
License
With the licence CC-BY, authors retain the copyright, allowing anyone to download, reuse, re-print, modify, distribute, and/or copy their contribution. The work must be properly attributed to its author.
It is not necessary to ask for further permission from the author or journal board.
This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.