On identities involving the sixth order mock theta functions

Lovejoy, Jeremy

doi:10.1090/S0002-9939-10-10296-2

On identities involving the sixth order mock theta functions
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by Jeremy Lovejoy

Proc. Amer. Math. Soc. 138 (2010), 2547-2552

DOI: https://doi.org/10.1090/S0002-9939-10-10296-2

Published electronically: March 3, 2010

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Abstract:

We present $q$-series proofs of four identities involving sixth order mock theta functions from Ramanujan’s lost notebook. We also show how Ramanujan’s identities can be used to give a quick proof of four sixth order identities of Berndt and Chan.

References

G. E. Andrews, On basic hypergeometric series, mock theta functions, and partitions. I, Quart. J. Math. Oxford Ser. (2) 17 (1966), 64–80. MR 193282, DOI 10.1093/qmath/17.1.64
George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part II, Springer, New York, 2009. MR 2474043
George E. Andrews and Dean Hickerson, Ramanujan’s “lost” notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60–105. MR 1123099, DOI 10.1016/0001-8708(91)90083-J
Bruce C. Berndt and Song Heng Chan, Sixth order mock theta functions, Adv. Math. 216 (2007), no. 2, 771–786. MR 2351377, DOI 10.1016/j.aim.2007.06.004
J. M. Borwein, P. B. Borwein, and F. G. Garvan, Some cubic modular identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), no. 1, 35–47. MR 1243610, DOI 10.1090/S0002-9947-1994-1243610-6
George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
L. Lorenz, Bidrag til tallenes theori, Tidsskrift for Mathematik (3) 1 (1871), 97-114.
Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735