Article ID: MTJPAM-D-21-00075

Title: Bilateral mock theta functions of order eleven and their Lerch representations


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-21-00075; Volume 4 / Issue 2 / Year 2022, Pages 60-71

Document Type: Research Paper

Author(s): Mohammad Ahmad a , Shahab Faruqi b

aDepartment of Mathematics, National Defence Academy, Khadakwasla, Pune, India

bDepartment of Mathematics, National Defence Academy, Khadakwasla, Pune, India

Received: 14 December 2021, Accepted: 1 June 2022, Published: 10 August 2022.

Corresponding Author: Mohammad Ahmad (Email address: mahmad_786@rediffmail.com)

Full Text: PDF

Abstract

We use bilateral basic hypergeometric series to obtain some bilateral mock theta functions and show that these functions are related to the basic hypergeometric series 6Φ5. Also Ramanujan’s characterization of mock theta functions is satisfied by these functions. We also express them in terms of the Lerch’s transcendental function f(x, ξ; q, p).

Keywords: Mock theta functions, bilateral mock theta functions, Lerch transcendent, hypergeometric series, characteristic property, unit circle

References:
  1. R. P. Agarwal, Mock theta functions – An analytical point of view, Proc. Nat. Acad. Sci. India 64 (A.I), 95–106, 1994.
  2. M. Ahmad, On the behavior of bilateral mock theta functions-I, In: Algebra and Analysis: Theory and Application (Ed. by N. M. Khan and M. Imdad), Narosa Publishing House, New Delhi, 259–273, 2015.
  3. M. Ahmad and S. Faruqi, Some bilateral mock theta functions and their Lerch representations, Aligarh Bull. Math. 34 (1-2), 75–92, 2015.
  4. M. Ahmad, S. Haq and A. H. Khan, Bilateral mock theta functions and further properties, Electron. J. Math. Anal. Appl. 7 (2), 216–229, 2019.
  5. G. E. Andrews and D. Hickerson, The sixth order mock theta functions, Adv. Math. 89, 60–105, 1991.
  6. G. E. Andrews, q-orthogonal polynomials, Roger-Ramanujan identities and mock theta functions, Proc. Steklov Inst. Math. 276, 21–32, 2012.
  7. B. C. Berndt and S. H. Chan, Sixth order mock theta functions, Adv. Math. 216 (2), 771–786, 2007.
  8. K. Bringmann, K. Hikami and J. Lovejoy, On the modularity of the unified WRT invariants of certain Seifert manifolds, Adv. Appl. Math. 46 (1), 86–93, 2011.
  9. Y. S. Choi, Tenth order mock theta functions in Ramanujan’s lost note book, Invent. Math. 136, 497–569, 1999.
  10. L. A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. Amer. Math. Soc. 72, 474–500, 1952.
  11. N. J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Mono Graphs, Number 27, American Mathematical Society, Providence, Rhode, Island, 1988.
  12. G. Gasper and M. Rahman, Basic hypergeometric series, 2nd Ed., Cambridge University Press, Cambridge (UK), 2004.
  13. B. Gordon and R. J. McIntosh, Some eight order mock theta functions, J. Lond. Math. Soc. 62 (2), 321–335, 2000.
  14. A. Gupta, On certain Ramanujan’s mock theta functions, Proc. Indian Acad. Sci. 103, 257–267, 1993.
  15. M. Griffin, K. Ono and L. Rolen, Ramanujan’s mock theta functions, Proc. Natl. Acad. Sci. USA 110 (15), 5765–5768, 2013.
  16. K. Hikami, Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10, 509–530, 2005.
  17. K. Hikami, Transformation formulae of the 2nd order mock theta function, Lett. Math. Phys. 75 (1), 93–98, 2006.
  18. M. Lerch, Nov analogie rady theta a nekte zvltn hypergeometrick rady Heineovy, Rozpravy 3, 1–10, 1893.
  19. R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2), 284–290, 2007.
  20. S. D. Prasad, Certain extended mock theta functions and generalized basic hypergeometric transformation, Math. Scand. 27, 237–244, 1970.
  21. S. Ramanujan, Collected papers, Cambridge University Press, 1927 (Reprinted by Chelsea New York, 1960).
  22. R. C. Rhoades, On Ramanujan’s definition of mock theta function, Proc. Natl. Acad. Sci. USA 110 (19), 7592–7594, 2013.
  23. D. P. Shukla and M. Ahmad, Bilateral mock theta functions of order seven, Math Sci. Res. J. 7 (1), 8–15, 2003.
  24. D. P. Shukla and M. Ahmad, On the behaviour of bilateral mock theta functions of order seven, Math Sci. Res. J. 7 (1), 16–25, 2003.
  25. D. P. Shukla and M. Ahmad, Bilateral mock theta functions of order thirteen, Proc. Jangjeon Math. Soc. 6 (2), 167–183, 2003.
  26. B. Srivastava, Certain bilateral basic hypergeometric transformations and mock theta functions, Hiroshima Math. J. 29, 19–26, 1999.
  27. B. Srivastava, A study of bilateral forms of the mock theta functions of order eight, J. Chungcheong Math. Soc. 18 (2), 117–129, 2005.
  28. B. Srivastava, A mock theta function of second order, Int. J. Math. Math. Sci. 2009, 2019; Article ID: 978425.
  29. B. Srivastava, A study of bilateral new mock theta functions, American Journal of Mathematics and Statistics 2 (4), 64–69, 2012.
  30. G. N. Watson, The final problem: An account of the mock theta functions, J. Lond. Math. Soc. 11, 55–80, 1936.
  31. G. N. Watson, The mock theta functions (2), Proc. Lond. Math. Soc. 42, 274–304, 1937.