# A short note on higher Mordell integrals

## TL;DR

This paper explores the connection between higher depth quantum modular forms and higher Mordell integrals, extending the classical relationships known for mock modular forms using techniques from Bringmann, Kaszian, and Milas.

## Contribution

It demonstrates that double Eichler integrals of certain depth two quantum modular forms can be expressed as higher Mordell integrals, extending the understanding of their structure.

## Key findings

- Double Eichler integrals relate to higher Mordell integrals.
- Higher depth quantum modular forms can be expressed via double integrals.
- Extension of classical Mordell integral relationships to higher depth forms.

## Abstract

Classical mock modular and quantum modular forms are known to have an intimate relationship with Mordell integrals thanks to Zwegers' groundbreaking PhD thesis. More recently, generalisations of mock/quantum modular forms to so-called "higher depth" versions have been intensively studied. In essence, a mock/quantum modular form of depth $d$ is such that the error of modularity transforms as another mock/quantum modular form of depth $d - 1$. In this short note we use techniques of Bringmann, Kaszian, and Milas to show that the double Eichler integrals of a family of depth two quantum modular forms of weight one previously studied by the author can be related to certain "higher" Mordell integrals, meaning it may be written as a certain double integral, a la Zwegers.