Sabuj, Das and Mohajan, Haradhan (2014): Development of Partition Functions of Ramanujan’s Works. Published in: Journal of Environmental Treatment Techniques , Vol. 2, No. 4 (30 October 2014): pp. 143149.

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Abstract
In 1986, Dyson defined the rank of a partition as the largest part of a partition minus the number of parts of . In 1988, Garvan discussed the theta series in x like A(x), B(x), C(x), D(x) and also discussed Jacobi’s triple product Identity (1829). Both of the authors have worked on Ramanujan’s seminal works “Ramanujan’s Lost Notebooks”. This paper proves the Theorem 1 with the help of Dyson’s rank conjectures N(0,5,5n +1), N(2,5, 5n +1) and proves the Theorem 2 with the help of Garvan’s theta series and Dyson’s rank conjectures N(1,5, 5n+2), N(2,5, 5n+2), respectively. An attempt has been taken here to the development of the Ramanujan’s works with the contributions of Dyson and Garvan. Definitions and simple mathematical calculations are presented here to make the paper easier to the common readers.
Item Type:  MPRA Paper 

Original Title:  Development of Partition Functions of Ramanujan’s Works 
English Title:  Development of Partition Functions of Ramanujan’s Works 
Language:  English 
Keywords:  Congruent, Jacobi’s Triple product, Modulo, Ramanujan’s Lost Notebook, Theta series. 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C30  General 
Item ID:  83045 
Depositing User:  Haradhan Kumar Mohajan 
Date Deposited:  01 Dec 2017 08:11 
Last Modified:  14 Oct 2019 16:31 
References:  [1] Andrews, G.E. and Garvan, F.G., Ramanuj’s Lost Notebook VI: The Mock Theta Conjectures, Advances in Math., 1989. 73: 242–255. [2] Garvan, F.G., Generalizations of Dyson’s Rank, Ph. D. Thesis, Pennsylvania State University, 1986. [3] Garvan, F.G. Partitions Yesterday and Today, New Zealand Math. Soc., Wellington, 1979. [4] Garvan, F.G. Combinatorial Interpretations of Ramanaujan’s Partition Congruences, Trans. Amer. Math. Soc. 1988. 305: 47–77. [5] Das, S. and Mohajan, H.K., Generating Funtions for P(n,p,*),and P(n,*,p), Amer. Rev. of Math. and Sta. 2014. 2(1): 33–36. [6] Das, S. and Mohajan, H.K., Mock Theta Conjectures, Jour. of Env. Treat. Tech. 2014, 2(1): 22–28. [7] Atkin, A.O.L., Proof of a Conjecture of Ramanujan, Glasgow Math. J. 1967. 8: 14–32. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/83045 