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Transformation Formulas: 10th Order Mock Theta Functions

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Abstract

On page 9 in his Lost Notebook [232], Ramanujan offers eight identities for tenth order mock theta functions.

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References

  1. G.E. Andrews and D. Hickerson, Ramanujan’s “Lost Notebook” VII: The sixth order mock theta functions, Adv. Math. 89 (1991), 60–105.

    Article  MathSciNet  Google Scholar 

  2. B.C. Berndt, Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.

    Book  Google Scholar 

  3. B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.

    Google Scholar 

  4. B.C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.

    Google Scholar 

  5. B.C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, American Mathematical Society, Providence, RI, 1995; London Mathematical Society, London, 1995.

    Google Scholar 

  6. Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook, Invent. Math. 136 (1999), 497–569.

    Article  MathSciNet  Google Scholar 

  7. Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook. II, Adv. Math. 156 (2000), 180–285.

    Google Scholar 

  8. Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook. IV, Trans. Amer. Math. Soc. 354 (2002), 705–733.

    Google Scholar 

  9. Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24 (2011), 345–386.

    Article  MathSciNet  Google Scholar 

  10. G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd. ed., Cambridge University Press, Cambridge, 2004.

    Book  Google Scholar 

  11. B. Gordon and R.J. McIntosh, Some eighth order mock theta functions, J. London Math. Soc. (2) 62 (2000), 321–335.

    Google Scholar 

  12. B. Gordon and R.J. McIntosh, Modular transformations of Ramanujan’s fifth and seventh order mock theta functions, Ramanujan J. 7 (2003), 193–222.

    Article  MathSciNet  Google Scholar 

  13. B. Gordon and R.J. McIntosh, A survey of classical mock theta functions, in Partitions, q-Series and Modular Forms, K. Alladi and F. Garvan, eds., Develop. in Math. 23, 2011, Springer, New York, pp. 95–144.

    Google Scholar 

  14. D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639–660.

    Article  MathSciNet  Google Scholar 

  15. D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), 661–677.

    Article  MathSciNet  Google Scholar 

  16. M.I. Knopp, Modular Functions in Analytic Number Theory, Markham, Chicago, 1970; reprinted by Chelsea, New York, 1993; reprinted by the American Mathematical Society, Providence, RI.

    Google Scholar 

  17. T.H. Koornwinder, On the equivalence of two fundamental theta identities, Anal. Applics. 12 (2014), 711–725.

    Article  MathSciNet  Google Scholar 

  18. R.J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2) (2007), 284–290.

    Article  MathSciNet  Google Scholar 

  19. R.J. McIntosh, New mock theta conjectures, Part I, Ramanujan J. 46 (2018), 593–604.

    Article  MathSciNet  Google Scholar 

  20. L.J. Mordell, The value of the definite integral \(\int _{-\infty }^{\infty }{e^{at^{2}+bt } \over e^{ct} + d}dt\), Quart. J. Math. 48 (1920), 329–342.

    MATH  Google Scholar 

  21. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

    Google Scholar 

  22. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

    Google Scholar 

  23. R.A. Rankin, Modular Forms and Functions, Cambridge University Press, Cambridge, 1977.

    Book  Google Scholar 

  24. S. Robins, Generalized Dedekind η-products, Contemp. Math. 166 (1994), 119–128.

    Article  MathSciNet  Google Scholar 

  25. G.N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55–80; reprinted in [68], pp. 325–347.

    Google Scholar 

  26. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.

    MATH  Google Scholar 

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Andrews, G.E., Berndt, B.C. (2018). Transformation Formulas: 10th Order Mock Theta Functions. In: Ramanujan's Lost Notebook. Springer, Cham. https://doi.org/10.1007/978-3-319-77834-1_12

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