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Cubic theta functions. (English) Zbl 1107.33022

Summary: Some new identities for the four cubic theta functions \(a'(q,z)\), \(a(q,z)\), \(b(q,z)\) and \(c(q,z)\) are given. For example, we show that \[ a'(q,z)^{3}= b(q,z)^{3}+c(q)^{2}c(q,z). \] This is a counterpart of the identity \[ a(q,z)^{3}=b(q)^{2}b(q,z^{3})+c(q,z)^{3}, \] which was found by M. D. Hirschhorn, F. Garvan and J. Borwein [Can. J. Math. 45, No. 4, 673–694 (1993; Zbl 0797.33012)].
The Laurent series expansions of the four cubic theta functions are given. Their transformation properties are established using an elementary approach due to K. Venkatachaliengar [Development of elliptic functions according to Ramanujan. Madurai: Madurai Kamaraj University, Department of Mathematics (1988; Zbl 0913.33012), Hackensack, NJ: World Scientific (2012; Zbl 1247.11003)]. By applying the modular transformation to the identities given by Hirschhorn et al., several new identities in which \(a'(q,z)\) plays the role of \(a(q,z)\) are obtained

MSC:

33E05 Elliptic functions and integrals
05A30 \(q\)-calculus and related topics
11B65 Binomial coefficients; factorials; \(q\)-identities
11F20 Dedekind eta function, Dedekind sums
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

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