A family of vector-valued quantum modular forms of depth two

TL;DR

This paper introduces a new family of vector-valued quantum modular forms of depth two, exploring their properties, representations as Eichler integrals and theta functions, and their role in indefinite theta series.

Contribution

It presents a novel family of quantum modular forms with explicit constructions and connections to double Eichler integrals and non-holomorphic theta functions.

Findings

Established quantum modular properties of the new functions

Represented functions as double Eichler integrals and non-holomorphic theta functions

Connected these functions to weight two indefinite theta series

Abstract

We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.