A short note on higher Mordell integrals

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.