Qiuxia HU

Some new representations of Hikami’s second-order mock theta function D5(q)

In this paper, a second-order mock theta function D5(q) given by Hikami [11] is studied. By using basic hypergeometric transformation formulae, we attain some new representations of Hikami’s mock theta function D5(q). Meanwhile, dual nature of bilateral series associated to mock theta function D5(q) is also discussed.

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1] Adiga C, Bulkhali NAS, Simsek Y, Srivastava HM. A continued fraction of Ramanujan and some Ramanujan-Weber class invariants. Filomat 2017; 13 (31): 3975-3997.
[2] Arjika S. Certain generating functions for Cigler’s polynomials, Montes Taurus Journal of Pure and Applied Mathematics 2021; 3 (3): 284-296.
[3] Andrews GE, Hickerson D. Ramanujan’s “lost” notebook VII: The sixth order mock theta functions. Advances in Mathematics 1991; 89 (1): 60-105.
[4] Chen B. On the dual nature theory of bilateral series associated to mock theta functions. International Journal of Number Theory 2018; 14: 63–94.
[5] Choi YS. The basic bilateral hypergeometric series and the mock theta functions. Ramanujan Journal 2011; 24 (3): 345-386.
[6] Choi J, K Rathie A. General summation formulas for the Kamp De Friet function. Montes Taurus Journal of Pure and Applied Mathematics 2019; 1 (1): 107-128.
[7] Gasper G, Rahman M. Basic hypergeometric series, 2nd edition. Cambridge University Press, Cambridge, 2004.
[8] Gordon B, McIntosh RJ. A survey of classic mock theta functions. In Partitions, q-Series, and Modular Forms, Developments in Mathematics 2012; 23: 95-144.
[9] Hickerson DR. A proof of the mock theta conjectures. Inventiones Mathematicae 1988; 94 (3): 639-660.
[10] Hickerson DR and Mortenson ET. Hecke-type double sums, Appell-Lerch sums and mock theta functions, I. Proceedings of the London Mathematical Society 2014; 109 (2): 382-422.
[11] Hikami K. Mock (false) theta functions as quantum invariants. Regular and Chaotic Dynamics 2005; 10 (4): 509-530.
[12] Hikami K. Transformation formula of the “second” order Mock theta function. Letters in Mathematical Physics 2006; 75 (1): 93-98.
[13] Hu QX, Song HF, Zhang ZZ. Third-order mock theta functions. International Journal of Number Theory 2020; 16 (1): 91-106.
[14] Kacand V, Cheung P. Quantum calculus. Springer, New York, 2002.
[15] Koepf W. Hypergeometric Summation: An algorithmic approach to summation and special function identities. Braunschweig, Germany: Vieweg 1998; pp.25-26.
[16] McIntosh RJ. Second order mock theta functions. Canadian Mathematical Bulletin 2007; 50 (2): 284-290.
[17] Mc Laughlin J. Mock theta functions identities deriving from bilateral basic hypergeometric series. Springer Proceedings in Mathematics and Statistics 2017; 221: 503-531.
[18] Mortenson ET. On the dual nature of partial theta functions and Appell-Lerch sums. Advances in Mathematics 2014; 264 :236-260.
[19] Slater LJ. Generalized hypergeometric functions. Cambridge Univ. Press, Cambridge, London, New York, 1966.
[20] Srivastava B. A mock theta function of second order. International Journal of Mathematics and Mathematical Sciences, Volume 2009, Article ID 978425, 15 pages.
[21] Srivastava HM, Chaudhary MP, Wakene FK. A family of theta-function identities based upon q -binomial theorem and Heine’s Transformations. Montes Taurus Journal of Pure and Applied Mathematics 2020; 2 (2): 16.
[22] Srivastava HM, Karlsson PW. Multiple gaussian hypergeometric series, Halsted Ellis Horwood, Chichester; Wiley, New York, 1985.
[23] Watson GN. The final problem: an account of the mock theta functions. The Journal of the London Mathematical Society 1936; 11 (1): 55-80.
[24] Zwegers S. Mock Theta Functions. PhD Thesis (2002), Univ of Utrecht, The Netherlands.