Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2023-10-03 , DOI: 10.1007/s41980-023-00817-0 Su-Ping Cui , Nancy S. S. Gu , Dazhao Tang
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.
中文翻译：
模拟 Theta 函数和 Appell-Lerch 和的推广
拉马努金命名并首先研究了模拟 theta 函数，这些函数可以用欧拉形式、Appell-Lerch 和、Hecke 型二重和以及亚纯雅可比形式的傅立叶系数来表示。在本文中，我们研究了模拟 theta 函数的一些概括，并用 Appell-Lerch 和来表达它们。例如，本文证明的一个结果是，对于任何正整数r、\(|q|<1\)和x，没有分母消失
$$\begin{对齐}&\big (1+x^{-1}\big ) \sum _{n=0}^{\infty }\dfrac{(-q;q)_{2n+2r- 2}q^{n+1}}{(xq^{2r-1},x^{-1}q^{2r-1};q^2)_{n+1}}\\&\qquad =\dfrac{1}{(q,q,q^2;q^2)_\infty }\sum _{j=0}^{2r-2}q^{1-j} \dfrac{(q ^2;q^2)_{2r-2}}{(q^2;q^2)_j(q^2;q^2)_{2r-2-j}}\sum _{n=- \infty }^\infty \dfrac{(-1)^nq^{n(n+2)}}{1-xq^{2n+2r-2j-1}}。\end{对齐}$$此外，我们不仅概括了拉马努金的两个通用模拟theta函数\(g_2(x,q)\)和\(g_3(x,q)\)，而且概括了拉马努金在他丢失的笔记本中记录的两个身份。
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