Abstract
Mock theta functions have been deeply studied in the literature. Historically, there are many forms of representations for mock theta functions: Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, we first establish Hecke-type double sums for the second and eighth order mock theta functions by Bailey’s lemma and a Bailey pair given by Andrews and Hickerson. Meanwhile, we give different proofs of the generalized Lambert series for the mock theta functions A(q), \(U_0(q)\), and \(U_1(q)\). Furthermore, using Ramanujan’s \({_1}\psi _1\) summation formula and a \({_2}\psi _2\) transformation formula due to Bailey, we prove some identities related to these mock theta functions.
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We would like to thank the referee for valuable suggestions.
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This work was supported by the National Natural Science Foundation of China and the Fundamental Research Funds for the Central Universities of China.
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Cui, SP., Gu, N.S.S. & Hao, LJ. On second and eighth order mock theta functions. Ramanujan J 50, 393–422 (2019). https://doi.org/10.1007/s11139-018-0045-4
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DOI: https://doi.org/10.1007/s11139-018-0045-4