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Development of Partition Functions of Ramanujan’s Works

Author

Listed:
  • Sabuj, Das
  • Mohajan, Haradhan

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.

Suggested Citation

  • Sabuj, Das & Mohajan, Haradhan, 2014. "Development of Partition Functions of Ramanujan’s Works," MPRA Paper 83045, University Library of Munich, Germany, revised 15 Aug 2014.
    • Handle: RePEc:pra:mprapa:83045
    as

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    File URL: https://mpra.ub.uni-muenchen.de/83045/1/MPRA_paper_83045.pdf
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    References listed on IDEAS

    as
    1. Das, Sabuj & Mohajan, Haradhan, 2014. "Mock Theta Conjectures," MPRA Paper 55821, University Library of Munich, Germany, revised 04 Apr 2014.
    2. Das, Sabuj & Mohajan, Haradhan, 2014. "Generating Functions for and," MPRA Paper 55684, University Library of Munich, Germany, revised 22 Jan 2014.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Mohajan, Devajit & Mohajan, Haradhan, 2023. "Economic Aspects of Profit Maximization if Cost of Principal Raw Material Increases," MPRA Paper 117453, University Library of Munich, Germany, revised 22 Mar 2023.
    2. Mohajan, Devajit & Mohajan, Haradhan, 2023. "The Responses of an Organization for the Increase in Wage Rates: Profit Maximization Cases," MPRA Paper 118238, University Library of Munich, Germany, revised 10 Jun 2023.
    3. Mohajan, Devajit & Mohajan, Haradhan, 2023. "Economic Investigation of Lagrange Multiplier if Cost of Inputs and Budget Size of a Firm Increase: A Profit Maximization Endeavor," MPRA Paper 117993, University Library of Munich, Germany, revised 07 May 2023.
    4. Mohajan, Devajit & Mohajan, Haradhan, 2023. "Various Problems Arise in Industrial Economics If Wage Rate Increases: A Study for Nonlinear Budget Constraint," MPRA Paper 117553, University Library of Munich, Germany, revised 04 Apr 2023.
    5. Mohajan, Devajit & Mohajan, Haradhan, 2023. "Effects of Various Inputs for Increased Interest Rate of Capital: A Nonlinear Budget Constraint Consideration," MPRA Paper 118134, University Library of Munich, Germany, revised 06 May 2023.
    6. Mohajan, Devajit & Mohajan, Haradhan, 2023. "An Economical Study When Cost of Irregular Raw Materials of an Industry Increases for Nonlinear Budget Constraint," MPRA Paper 118176, University Library of Munich, Germany, revised 05 May 2023.

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    More about this item

    Keywords

    Congruent; Jacobi’s Triple product; Modulo; Ramanujan’s Lost Notebook; Theta series.;
    All these keywords.

    JEL classification:

    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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