Abstract
On page 202 in his Lost Notebook, Ramanujan recorded without proofs two modular transformations involving a Mordell integral, q-hypergeometric series, and generalized Lambert series. These two formulas were first proved by Y.-S. Choi [110], and in this chapter we relate his proofs.
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G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part III, Springer, New York, 2012.
B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
B. Chern and R.C. Rhoades, The Mordell integral, quantum modular forms, and mock Jacobi forms, Res. Number Theory 1 (2015), 14 pp.
Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook. IV, Trans. Amer. Math. Soc. 354 (2002), 705–733.
Y.-S. Choi, Generalization of two identities in Ramanujan’s lost notebook, Acta Arith. 114 (2004), 369–389.
Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24 (2011), 345–386.
Y.-S. Choi, Modular transformations involving the Mordell integral in Ramanujan’s lost notebook, Pacific J. Math. 272 (2014), 59–85.
F.G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77.
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.
D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639–660.
L. Kronecker, Summierung der Gausschen Reihen \(\sum _{h=0}^{h=n-1}e^{2h^{2}\pi i/n }\), J. Reine Angew. Math. 105 (1889), 267–268.
L. Kronecker, Zur Dastellung von Reihen durch Integrale, J. Reine Angew. Math. 105 (1889), 345–354.
J. Lovejoy and R. Osburn, On two 10th order mock theta functions, Ramanujan J. 36 (2015), 117–121.
R.J. McIntosh, On the universal mock theta function g 2 and Zwegers’ μ-function, Analytic Number Theory, Modular Forms and q-Hypergeometric Series (in Honor of Krishna Alladi’s 60th Birthday, University of Florida, Gainesville, March 2016), G.E. Andrews and F. Garvan, eds., Springer, 2018, pp. 497–502.
W. Moore, Modular transformations of Ramanujan’s tenth order mock theta functions, preprint.
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.
L.J. Mordell, The definite integral \(\int _{-\infty }^{\infty }{e^{at^{2}+bt } \over e^{ct} + d}dt\) and the analytic theory of numbers, Acta Math. 61 (1933), 323–360.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
C.L. Siegel, Über Riemanns Nachlass zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2 (1932), 45–80.
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.
S. Zwegers, Mock Theta Functions, Doctoral Dissertation, Universiteit Utrecht, 2002.
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Andrews, G.E., Berndt, B.C. (2018). Two Identities Involving a Mordell Integral and Appell–Lerch Sums. In: Ramanujan's Lost Notebook. Springer, Cham. https://doi.org/10.1007/978-3-319-77834-1_13
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