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On certain Ramanujan's mock theta functions

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Abstract

Ramanujan's last gift to the mathematicians was his ingeneous discovery of the mock theta functions of order three, five and seven. Recently, Andrews and Hickerson found a set of seven more functions in Ramanujan's Lost Note Book and formally labelled them as mock theta functions of order six. In this paper the complete forms of these functions have been studied and connected with the bilateral basic hypergeometric series2Ψ2. Several other interesting properties and transformations have also been studied.

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References

  1. Agarwal R P, Certain basic hypergeometric identities associated with mock theta functions,Q. J. Math. (Oxford)20 (1968) 121–128

    Article  Google Scholar 

  2. Agarwal R P, A ‘Lost’ notebook of Ramanujan of G E Andrews,Adv. Math. (M.I.T.) 53 (1984) 291–300

    Article  MATH  Google Scholar 

  3. Agarwal R P, Ramanujan's Last Gift,Math. Stud. 58 (1991) 121–150

    Google Scholar 

  4. Agarwal R P, Lambert series and Ramanujan,Platinum Jubliee Lecture delivered at the Indian Science Congress session (1992)at Baroda. (under publication)

  5. Agarwal R P, Mock theta functions—An analytical point of viewNat. Acad. Sci. Letters. 15 (1992) 51–52 Detailed paper under publication inProc. Nat. Acad. Sci. India (1993)

    MATH  Google Scholar 

  6. Andrews G E, On basic hypergeometric series, mock theta functions and partitions—I,Q. J. Math. (Oxford)17 (1966) 64–80

    Article  MATH  Google Scholar 

  7. Andrews G E, On basic hypergeometric series, mock theta functions and partitions—II,Q. J. Math. (Oxford)17 (1966) pp. 132–143

    Article  MATH  Google Scholar 

  8. Andrews G E, On the theorems of Watson and Dragonette for Ramanujan's mock theta functions,Am. J. Math. 88 (1966) 454–90

    Article  Google Scholar 

  9. Andrews G E, An introduction to Ramanujan's ‘Lost’ Notebook,Am. Math. Mon. 86 (1979) 89–108

    Article  MATH  Google Scholar 

  10. Andrews G E, The fifth and seventh order mock theta functions,Trans. Am. Math. Soc. 293 (1986) 113–134

    Article  MATH  Google Scholar 

  11. Andrews G E, Ramanujan's fifth order mock theta functions,Ramanujan revisited.Acad. Press, N. Y. (1988) 47–56

    Google Scholar 

  12. Andrews G E, Ramanujan's ‘Lost’ notebook-I Partial θ-functions,Adv. Math. 2,41 (1981) 137–172

    Article  MATH  Google Scholar 

  13. Andrews G E and Garvan F G, Ramanujan's ‘Lost’ Notebook-VI—The mock theta conjectures,Adv. Math. 73 (1989) 242–255

    Article  MATH  MathSciNet  Google Scholar 

  14. Andrews G E and Hickerson Dick, Ramanujan's ‘Lost’ Notebook VII: The sixth order mock theta functions,Adv. Math. 89 (1991) 60–104

    Article  MATH  MathSciNet  Google Scholar 

  15. Bailey W N, On the basic bilateral hypergeometric series2Ψ2,Q. J. Math. (Oxford)2 (1950) 194–198

    Article  MathSciNet  Google Scholar 

  16. Fine N J,Basic hypergeometric series and Applications (1988) Mathematical Surverys and Monographs, Vol 27,Amer. Math. Soc. Providence, RI

    MATH  Google Scholar 

  17. Gupta Mukul, Certain basic hypergeometric transformations, mock theta functions and related identities—I,Proc. Natl. Acad. Sci., India. 59 (1989) 573–586

    MATH  Google Scholar 

  18. ibid II,60 (1990) 297–305

    MATH  Google Scholar 

  19. Gupta A, A study of transformations for a special type of3Φ2 series and Mock theta functions (under publication inGanita 44 (1993))

  20. Gupta A, A study of certain Basic Hypergeometric Functions and Transformations Associated with Ramanujan, (Doctoral thesis approved by the University of Lucknow) (1992)

  21. Ramanujan S,The ‘Lost’ Notebook and other unpulished papers. (1988) Narosa Publishing House, New Delhi

    Google Scholar 

  22. Sears D B, On the transformation theory of Basic Hypergeometric function,Proc. London Math. Soc. 53 (1951) 158–179

    Article  MATH  MathSciNet  Google Scholar 

  23. Slater L J, Further identities of the Rogers-Ramanujan type,Proc. Lond. Math. Soc., (2)54 (1952) 147–167

    Article  MATH  MathSciNet  Google Scholar 

  24. Slater L JGeneralized Hypergeometric series. (1966) Camb. Uni. Press

  25. Watson G N, The final problem: an account of the mock theta functions,J. London Math. Soc. 11 (1936) 55–80

    Article  MATH  Google Scholar 

  26. Watson G N, The mock theta functions (2),Proc. Lond. Math. Soc. 42 (1937) 274–304

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Astronomy, University of Lucknow, 226 007, Lucknow, India

    Anju Gupta

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  1. Anju Gupta
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Gupta, A. On certain Ramanujan's mock theta functions. Proc. Indian Acad. Sci. (Math. Sci.) 103, 257–267 (1993). https://doi.org/10.1007/BF02866990

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  • DOI: https://doi.org/10.1007/BF02866990

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