On a relation between certain $q$-hypergeometric series and Maass waveforms

TL;DR

This paper explores the connection between specific $q$-hypergeometric series and Maass waveforms, revealing that these series interpolate coefficients of certain Maass waveforms, thus bridging special functions and automorphic forms.

Contribution

It establishes a new link between $q$-series from Ramanujan's notebook and Maass waveforms, answering a question posed by Li, Ngo, and Rhoades.

Findings

$q$-series $\sigma$ and $\sigma^*$ interpolate Maass waveform coefficients

Results extend Cohen's theorem relating $q$-series to automorphic forms

Provides new insights into the structure of $q$-hypergeometric series and Maass waveforms

Abstract

In this paper, we answer a question of Li, Ngo, and Rhoades concerning a set of $q$-series related to the $q$-hypergeometric series $\sigma$ from Ramanajun's lost notebook. Our results parallel a theorem of Cohen which says that $\sigma$, along with its partner function $\sigma^*$, interpolate the coefficients of a Maass waveform of eigenvalue $1/4$.