Whittaker Fourier type solutions to differential equations arising from string theory

In this article, we find the full Fourier expansion for solutions of $(\Delta-\lambda)f(z) = -E_k (z) E_\ell (z)$ for $z = x + i y \in \mathfrak{H}$ for certain values of parameters $k$, $\ell$ and $\lambda$. When such an $f$ is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory.