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PARTIAL INDEFINITE THETA IDENTITIES

Part of: Basic hypergeometric functions Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  19 September 2016

BYUNGCHAN KIM  and
JEREMY LOVEJOY
BYUNGCHAN KIM
Affiliation:
School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, Korea email bkim4@seoultech.ac.kr
JEREMY LOVEJOY*
Affiliation:
CNRS, LIAFA, Université Denis Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France email lovejoy@math.cnrs.fr
*
✉ lovejoy@math.cnrs.fr
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Abstract

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Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

Keywords

partial theta functionsindefinite theta functionspartial indefinite theta functionsBailey pairs

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F37: Forms of half-integer weight; nonholomorphic modular forms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 2 , April 2017 , pp. 255 - 289
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

This research was supported by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) of Korea (NRF-2014K1A3A1A21000358), and the STAR program number 32142ZM.

References

Andrews, G. E., ‘Ramanujan’s “lost” notebook. I. Partial 𝜃-functions’, Adv. Math. 41(2) (1981), 137–172.CrossRefGoogle Scholar
Andrews, G. E., ‘Hecke modular forms and the Kac-Peterson identities’, Trans. Amer. Math. Soc. 283(2) (1984), 451–458.Google Scholar
Andrews, G. E., ‘The fifth and seventh order mock theta functions’, Trans. Amer. Math. Soc. 293 (1986), 113–134.CrossRefGoogle Scholar
Andrews, G. E., ‘Bailey’s transform, lemma, chains and tree’, in: Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Science Series II: Mathematics, Physics and Chemistry, 30 (Kluwer Academic Publishers, Dordrecht, 2001), 1–22.Google Scholar
Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook. Part II (Springer, New York, 2009).Google Scholar
Andrews, G. E. and Hickerson, D., ‘The sixth order mock theta functions’, Adv. Math. 89 (1991), 60–105.Google Scholar
Bringmannm, K., Rolen, L. and Zwegers, S., ‘On the modularity of certain functions from the Gromov–Witten theory of elliptic orbifolds’, R. Soc. Open Sci. 2(11) 150310 (2015).Google Scholar
Fine, N., Basic Hypergeometric Series and Applications (American Mathematical Society, Providence, RI, 1988).Google Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Hickerson, D. R. and Mortenson, E. T., ‘Hecke-type double sums, Appell–Lerch sums, and mock theta functions, I.’, Proc. Lond. Math. Soc. (3) 109 (2014), 382–422.CrossRefGoogle Scholar
Hikami, K. and Lovejoy, J., ‘Torus knots and quantum modular forms’, Res. Math. Sci. 2 (2015), 2.Google Scholar
Hirschhorn, M. D. and Sellers, J., ‘Elementary proofs of various facts about 3-cores’, Bull. Aust. Math. Soc. 79 (2009), 507–512.Google Scholar
Ji, K. and Zhao, A. X. H., ‘The Bailey transform and Hecke-Rogers identities for the universal mock theta functions’, Adv. Appl. Math. 65 (2015), 65–86.CrossRefGoogle Scholar
Kim, B. and Lovejoy, J., ‘The rank of a unimodal sequence and a partial theta identity of Ramanujan’, Int. J. Number Theory 10 (2014), 1081–1098.CrossRefGoogle Scholar
Kim, B. and Lovejoy, J., ‘Ramanujan-type partial theta identities and rank differences for special unimodal sequences’, Ann. Comb. 19 (2015), 705–733.CrossRefGoogle Scholar
Kostov, V. P., ‘Asymptotic expansions of zeros of a partial theta function’, C. R. Acad. Bulgare Sci. 68(4) (2015), 419–426.Google Scholar
Kostov, V. P., ‘On the spectrum of a partial theta function’, Proc. Roy. Soc. Edinburgh Sect. A 144(5) (2014), 925–933.CrossRefGoogle Scholar
Kostov, V. P., ‘A property of a partial theta function’, C. R. Acad. Bulgare Sci. 67(10) (2014), 1319–1326.Google Scholar
Kostov, V. P. and Shapiro, B., ‘Hardy-Petrovitch-Hutchinson’s problem and partial theta function’, Duke Math. J. 162(5) (2013), 825–861.CrossRefGoogle Scholar
Lovejoy, J., ‘Ramanujan-type partial theta identities and conjugate Bailey pairs’, Ramanujan J. 29 (2012), 51–67.CrossRefGoogle Scholar
Lovejoy, J., ‘Bailey pairs and indefinite quadratic forms’, J. Math. Anal. Appl. 410 (2014), 1002–1013.Google Scholar
Mortenson, E., ‘On three third order mock theta functions and Hecke-type double sums’, Ramanujan J. 30 (2013), 279–308.CrossRefGoogle Scholar
Prellberg, T., ‘The combinatorics of the leading root of the partial theta function’, Preprint, 2012, http://arxiv.org/abs/1210.0095.Google Scholar
Sokal, A. D., ‘The leading root of the partial theta function’, Adv. Math. 229(5) (2012), 2603–2621.CrossRefGoogle Scholar
Warnaar, S. O., ‘50 years of Bailey’s lemma’, in: Algebraic Combinatorics and Applications (Gößweinstein, 1999) (Springer, Berlin, 2001), 333–347.CrossRefGoogle Scholar
Warnaar, S. O., ‘Partial theta functions. I. Beyond the lost notebook’, Proc. Lond. Math. Soc. (3) 87 (2003), 363–395.Google Scholar
Westerholt-Raum, M., ‘H-harmonic Maaß–Jacobi forms of degree 1’, Res. Math. Sci. 2 (2015), 12.Google Scholar