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http://dx.doi.org/10.5666/KMJ.2016.56.1.173

A Note on Continued Fractions and Mock Theta Functions  

Srivastava, Pankaj (Department of Mathematics, Motilal Nehru National Institute of Technology)
Gupta, Priya (Department of Mathematics, Motilal Nehru National Institute of Technology)
Publication Information
Kyungpook Mathematical Journal / v.56, no.1, 2016 , pp. 173-184 More about this Journal
Abstract
Mock theta functions are the most interesting topic mentioned in Ramanujan's Lost Notebook, due to its emerging application in the field of Number theory, Quantum invariants theory and etc. In the present research articles we have made an attempt to develop continued fractions representation of all the existing Mock theta functions.
Keywords
Mock theta functions; Basic hypergeometric function; Continued fraction;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 A. K. Srivastava, On Partial Sums of Mock Theta Functions of Order Three, Proc. Indian Acad. Sci. (Math. Sci.), 107(1)(1997), 1-12.   DOI
2 A. K. Srivastava, Certain Continued Fraction Representations for Functions associated with Mock Theta Functions of Order Three, Kodai Mathematical Journal, 25(2002), 278-287.   DOI
3 Bhaskar Srivastava, Ramanujan's Mock Theta Functions, Math. J. Okayama Univ., 47(2005), 163-174.
4 Bhaskar Srivastava, A Comprehensive Study of Second Order Mock Theta Functions, Bull. Korean Math. Soc., 4(42)(2005), 889-900.
5 B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verleg New York, Inc. (1989).
6 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verleg New York, Inc.(1998).
7 B. C. Berndt and S. H. Chan, Sixth Order Mock Theta Functions, Advances in Mathematics, 216(2007), 771-786.   DOI
8 B. Gordon and R. J. McIntosh, Some Eight Order Mock Theta Functions, J. London Math. Soc., 62(2)(2000), 321-335.   DOI
9 G. E. Andrews and D. Hickerson, Ramanujan's Lost Notebook VII: The Sixth Order Mock Theta Functions, Adv. Math., 89(1991), 60-105.   DOI
10 G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebooks Part I, Springer-Verleg New York, Inc., (2005).
11 G. N. Watson, A final Problem: An Account of The Mock Theta Functions, J. London Math. Soc., 11(1936), 55-80.
12 H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Ltd., (1985).
13 H. M. Srivastava, Some Convolution Identities Based upon Ramanujan's Bilataral Sum, Bull. Austral. Math. Soc., 49(1994), 433-437.   DOI
14 H. M. Srivastava, Some Generalizations and Basic (or q-) Extensions of the Bernoulli, Euler and Genocchi Polynomials, Appl. Math. Inform. Sci., 5(2011), 390-444.
15 K. G. Ramanathan, Hypergeometric Series and Continued Fractions, Proc. of Indian Acad. Sci. (Math. Sci.), 97(1-3)(1987), 277-296.   DOI
16 K. Hikami, Mock (false) Theta Functions as Quantum Invariants, Regular and Chaotic Dynamics, 10(2005), 509-530.   DOI
17 K. Hikami, Transformation formula of the 2nd Order Mock Theta Functions, Lett. Math. Phys, 75(2006), 93-98.   DOI
18 L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cmbridge, London and New York, (1966).
19 Morris Kline, Mathematical thought from Ancient to Modern Time, Oxford Univ (1972) Press: 28-32.
20 M. Pathak and Pankaj Srivastava, A Note on Continued Fractions and $3{\psi}3$ Series, Italian J. Pure Appl. Math., 27(2010), 191-200.
21 Pankaj Srivastava, Certain Continued Fractions for quotients of two $3{\psi}3$ Series, Proc. Nat. Acad. Sci. India, 78(A)IV(2008), 327-330.
22 Pankaj Srivastava, Resonence of Continude Fractions Related to $2-\psi}2$ Basic Bilateral Hypergeometric Series, Kyungpook Math. J., 51(2011), 419-427.   DOI
23 Pankaj Srivastava and A. J. Wahidi, A Note on Hikami's Mock Theta Functions, Int. Journal of Math. Analysis, 5(43)(2011), 2103-2109.
24 Pankaj Srivastava and R. V. G. K. Mohan, Certain Flowers of Continued Fractions in The Garden of Generalized Lambert Series, Journal of Mathematics Research, 4(3)(2012), 36-43.
25 R. P. Agrawal, An Attempt Towards Presenting an Unified Theory for Mock Theta Functions, Proc. Int. Conf. SSFA, 1(2001), 11-19.
26 R. P. Agarwal, Mock Theta Functions-An Analytical Point of View, Proc. Nat. Acad. Sci. (India), 64(1994) , 95-107.
27 R. P. Agarwal, Resonance of Ramanujans Mathematics Vol. II, New Age International Pvt. Ltd., New Delhi (1995).
28 R. P. Agarwal, Resonance of Ramanujans Mathematics Vol. III, New Age International Pvt. Ltd., New Delhi (1998).
29 R. Y. Denis, On Generalization of Continued Fraction of Gauss, International J. Math. and Math. Sci., 13(4)(1990), 741-745.   DOI
30 R. Y. Denis, S. N. Singh and S. P. Singh, On Hypergeometric Functions and Ramanujan's Continued Fractions, The Indian Mathematical Society, (1907-2007)(2007), 25-50.
31 R. Y. Denis, S. N. Singh and S. P. Singh, On Certain Continued Fraction Representations of Poly-basic Series, South East Asian J. Math. Math. Sci., 8(2)(2010), 25-33.
32 S. Bhargava and C. Adiga, On Some Continued Fraction of Srinivas Ramanujan, Proc. Amer. Math. Soc., 92(1984), 13-18.   DOI
33 S. Bhargava, C. Adiga and D. D. Somashekara, On Certain Continued Fractions related to $3{\phi}2$ Basic Hypergeometric Functions, J. Math. Phys. Sci., 21(1987), 613-629.
34 S. N. Singh, Basic Hypergeometric Series and Continued Fractions, Math. Student, 56(1988), 91-96.
35 S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge; reprinted by Chelsea, New York, 1962; reprented by the American Mathematical Society, Providence, RI, 2000 (1927).
36 Y. S. Choi, Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook, Invent. Math., 136(1999), 497-569.   DOI
37 S. Ramanujan, Notebook of Ramanujan Vol. I and II, T. I. F. R., Bombay (1957).
38 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi (1988).
39 W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cmbridge, (1935), reprented by Stechert-Hafner, New York (1964).