Mock Maass theta functions (Q2907046)

Let \(\tau = x+iy \in \mathbb{H}\). Motivated by certain functions studied by Andrews-Dyson-Hickerson and Cohen, the author defines a class of indefinite theta functions \(\Phi_{a,b}^{c_1,c_2}(\tau)\) which are eigenfunctions of the weight \(0\) Laplace operator (with eigenvalue \(1/4\)). He completes the \(\Phi_{a,b}^{c_1,c_2}(\tau)\) to obtain a non-holomorphic modular form \(\widehat{\Phi}_{a,b}^{c_1,c_2}(\tau)\) of weight \(0\) satisfying NEWLINE\[NEWLINE y^{-3/2}\left(\Delta_0 - \frac{1}{4}\right)\widehat{\Phi}_{a,b}^{c_1,c_2} \in S_{3/2} \otimes \overline{S_{3/2}}, NEWLINE\]NEWLINE where \(S_k\) is the space of weight \(k\) cusp forms. In analogy with mock theta functions, he calls \(\Phi_{a,b}^{c_1,c_2}(\tau)\) a \textit{mock Maass theta function} and the function NEWLINE\[NEWLINE(\Delta_0 - \frac{1}{4})\widehat{\Phi}_{a,b}^{c_1,c_2}NEWLINE\]NEWLINE its \textit{shadow}.