Holomorphic projections and Ramanujan's mock theta functions

TL;DR

This paper develops a spectral method-based holomorphic projection operator for vector-valued harmonic weak Maass forms and modular forms, enabling new recursive formulas for Ramanujan's mock theta functions' Fourier coefficients.

Contribution

Introduces a novel holomorphic projection operator using spectral methods, facilitating the analysis of Ramanujan's mock theta functions.

Findings

Derived simple recursions for Fourier coefficients of mock theta functions

Established a new spectral approach for automorphic forms

Connected harmonic weak Maass forms with mock theta functions

Abstract

We employ spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.