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Generalizations of Mock Theta Functions and Appell–Lerch Sums

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Abstract

Ramanujan named and first studied mock theta functions which can be represented by Eulerian forms, Appell–Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, we investigate some generalizations of mock theta functions and express them in terms of Appell–Lerch sums. For instance, one result proved in the present paper is that for any positive integer r, \(|q|<1\), and x, so that no denominators vanish

$$\begin{aligned}&\big (1+x^{-1}\big ) \sum _{n=0}^{\infty }\dfrac{(-q;q)_{2n+2r-2}q^{n+1}}{(xq^{2r-1},x^{-1}q^{2r-1};q^2)_{n+1}}\\&\qquad =\dfrac{1}{(q,q,q^2;q^2)_\infty }\sum _{j=0}^{2r-2}q^{1-j} \dfrac{(q^2;q^2)_{2r-2}}{(q^2;q^2)_j(q^2;q^2)_{2r-2-j}}\sum _{n=-\infty }^\infty \dfrac{(-1)^nq^{n(n+2)}}{1-xq^{2n+2r-2j-1}}. \end{aligned}$$

In addition, we generalize not only two of Ramanujan’s universal mock theta functions \(g_2(x,q)\) and \(g_3(x,q)\), but also two identities recorded by Ramanujan in his lost notebook.

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Acknowledgements

The authors would like to express their sincere gratitude to two anonymous referees for their careful reading of the manuscript and many constructive suggestions, which improved the quality of the paper to a great extent. Su-Ping Cui was partially supported by the National Natural Science Foundation of China (Grant No. 12001309) and the Natural Science Foundation Youth Fund of Qinghai (Grant No. 2022-ZJ-972Q). Nancy S. S. Gu was partially supported by the National Natural Science Foundation of China (Grant No. 12171255). Dazhao Tang was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ–MSX0387), the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509), and the Doctoral Start-Up Research Foundation (No. 21XLB038) of Chongqing Normal University.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, Qinghai, People’s Republic of China

    Su-Ping Cui

  2. Academy of Plateau Science and Sustainability, Xining, 810008, Qinghai, People’s Republic of China

    Su-Ping Cui

  3. Center for Combinatorics, LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Nancy S. S. Gu

  4. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Dazhao Tang

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  1. Su-Ping Cui
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Correspondence to Dazhao Tang.

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Cui, SP., Gu, N.S.S. & Tang, D. Generalizations of Mock Theta Functions and Appell–Lerch Sums. Bull. Iran. Math. Soc. 49, 71 (2023). https://doi.org/10.1007/s41980-023-00817-0

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