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On some double-sum false theta series. (English) Zbl 1401.33012

False theta series are series which would be theta series if there was no alternation of the signs in some series terms. In this paper the authors show that some false theta functions describe functions known from the literature.
For example, let \(u(n)\) is the number of unimodal sequences of weight \(n\) such that the largest part in the partition before the peak is larger than or equal to the largest part in the partition after the peak. Then, by a theorem of B. Kim,
\[ \sum_{n\geq0}u(n)q^n=\frac{1}{(q)_\infty}g_{1,2,4}(q,-q^4,q). \]
The authors show that the function \(g_{1,2,4}(q,-q^4,q)\) is in fact a false theta series.
The similar study is performed for the sequence \(du(n)\), which counts the unimodal sequences with plateau.
Arithmetic properties of these sequences are also studied through their false theta representations.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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[1] Andrews, G. E.; Dyson, F.; Hickerson, D. R., Partitions and indefinite quadratic forms, Invent. Math., 91, 391-407 (1988) · Zbl 0642.10012
[2] Bringmann, K.; Kane, B., Multiplicative q-hypergeometric series arising from real quadratic fields, Trans. Amer. Math. Soc., 363, 2191-2209 (2011) · Zbl 1228.11157
[3] Cohen, H., \(q\)-Identities for Maass waveforms, Invent. Math., 91, 409-422 (1988) · Zbl 0642.10013
[4] Hickerson, D. R.; Mortenson, E. T., Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3), 109, 2, 382-422 (2014) · Zbl 1367.11047
[5] Kim, B., Group actions and partitions, Electron. J. Combin., 24, Article #P3.58 pp. (2017) · Zbl 1372.05014
[6] Kim, B.; Lovejoy, J., Ramanujan-type partial theta identities and rank differences for special unimodal sequences, Ann. Comb., 19, 4, 705-733 (2015) · Zbl 1326.05008
[7] Kim, B.; Lovejoy, J., Ramanujan-type partial theta identities and conjugate Bailey pairs, II. Multisums, Ramanujan J. (2018), in press · Zbl 1393.33018
[8] Kim, B.; Lim, S.; Lovejoy, J., Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity, Proc. Amer. Math. Soc., 144, 3687-3700 (2016) · Zbl 1404.11052
[9] Lovejoy, J., Overpartitions and real quadratic fields, J. Number Theory, 106, 178-186 (2004) · Zbl 1050.11085
[10] Lovejoy, J., Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J., 29, 51-67 (2012) · Zbl 1322.11107
[11] Lovejoy, J.; Osburn, R., Real quadratic double sums, Indag. Math. (N.S.), 26, 697-712 (2015) · Zbl 1320.33026
[12] Ramanujan, S., Notebooks (2 volumes) (1957), Tata Institute of Fundamental Research: Tata Institute of Fundamental Research Bombay
[13] Ramanujan, S., The Lost Notebook and Other Unpublished Papers (1988), Narosa: Narosa New Delhi · Zbl 0639.01023
[14] Rogers, L. J., On two theorems of combinatory analysis and some allied identities, Proc. Lond. Math. Soc., 16, 315-336 (1917) · JFM 46.0109.01
[15] Slater, L. J., A new proof of Rogers’s transformations of infinite series, Proc. Lond. Math. Soc. (2), 53, 460-475 (1951) · Zbl 0044.06102
[16] Warnaar, S. O., 50 years of Bailey’s lemma, (Betten, A.; etal., Algebraic Combinatorics and Applications (2001), Springer: Springer Berlin), 333-347 · Zbl 0972.11003
[17] Zagier, D., Quantum modular forms, (Quanta of Maths: Conference in Honor of Alain Connes. Quanta of Maths: Conference in Honor of Alain Connes, Clay Mathematics Proceedings, vol. 11 (2010), AMS and Clay Mathematics Institute), 659-675 · Zbl 1294.11084
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