An elementary alternative proof for Chan’s analogue of Ramanujan’s most beautiful identity and some inequality of the cubic partition

Koustav Banerjee and Prabir Das Adhikary
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 77—87
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Authors and affiliations

Koustav Banerjee
Ramakrishna Mission Vivekananda University
Belur, Howrah 711202, West Bengal, India

Prabir Das Adhikary
Pondicherry University
R. V. Nagar, Kalapet, Puducherry 605014, India

Abstract

In this paper we provide an alterative proof for the congruence modulo 3 of the cubic partition a(n). That apart we also examine inequalities for a(n) and provide upper bound for it in the fashion of the classic partition function p(n).

Keywords

  • Partitions
  • Cubic partitions
  • Partition congruences

AMS Classification

  • 11P84

References

  1. Chan, H.-C. (2010) Ramanujan’s cubic continued fraction and an analog of his most beautiful identity. International Journal of Number Theory, 6.03, 673–680.
  2. Kruyswijk, D. (1950) On some well-known properties of the partition function p(n) and Euler’s infinite product. Nieuw Arch. Wisk, 23, 97–107.
  3. Ahlgren, S. (2000) Distribution of the partition function modulo composite integers M. Mathematische Annalen, 318(4), 795–803.
  4. Chen,W.Y.C, & Lin, B. L. S. (2009) Congruences for the number of cubic partitions derived from modular forms. arXiv preprint arXiv:0910.1263.
  5. Berndt, B. C. (2006) Number theory in the spirit of Ramanujan (Vol. 34). American Mathematical Society.

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Cite this paper

APA

Banerjee, K. and Adhikary, P. D. (2017). An elementary alternative proof for Chan’s analogue of Ramanujan’s most beautiful identity and some inequality of the cubic partition. Notes on Number Theory and Discrete Mathematics, 23(1), 77-87.

Chicago

Banerjee, Koustav and Prabir Das Adhikary. “An Elementary Alternative Proof for Chan’s Analogue of Ramanujan’s Most Beautiful Identity and Some Inequality of the Cubic Partition.” Notes on Number Theory and Discrete Mathematics 23, no. 1 (2017): 77-87.

MLA

Banerjee, Koustav and Prabir Das Adhikary. “An Elementary Alternative Proof for Chan’s Analogue of Ramanujan’s Most Beautiful Identity and Some Inequality of the Cubic Partition.” Notes on Number Theory and Discrete Mathematics 23.1 (2017): 77-87. Print.

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