On some special families of $q$-hypergeometric Maass forms

TL;DR

This paper constructs new quantum modular forms from $q$-hypergeometric series related to torus knots, demonstrating their Maass cusp form properties using Habiro ring representations, and explores their deep mathematical connections.

Contribution

It introduces a novel method to establish the cuspidality of Maass waveforms via $q$-hypergeometric series and Habiro ring representations, linking quantum modular forms to Maass cusp forms.

Findings

Construction of new quantum modular forms from $q$-hypergeometric series.

Proof of cuspidality of associated Maass waveforms using Habiro ring.

Identification of connections between $q$-hypergeometric series and Maass waveforms.

Abstract

Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and their Fourier coefficients encode distinguished Maass cusp forms. The cuspidality of these Maass waveforms is proven by making use of the Habiro ring representations of the associated quantum modular forms. Thus, we provide an example of using the $q$-hypergeometric structure of associated series to establish modularity properties which are otherwise non-obvious. We conclude the paper with a number of motivating questions and possible connections with Hecke characters, combinatorics, and still mysterious relations between $q$-hypergeometric series and {the passage} from positive to negative coefficients of Maass waveforms.