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# Communications in Number Theory and Physics

## Volume 17 (2023)

### Number 3

### Whittaker Fourier type solutions to differential equations arising from string theory

Pages: 583 – 641

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n3.a2

#### Authors

#### Abstract

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.

K.K-L. acknowledges support from NSF Grant number DMS-2001909.

Received 23 September 2022

Accepted 16 May 2023

Published 7 November 2023