Transformation properties of Andrews-Beck $NT$ functions and generalized Appell-Lerch series

In 2021, Andrews mentioned that George Beck introduced a partition statistic $NT(r,m,n)$ which is related to Dyson's rank statistic. Motivated by Andrews's work, scholars have established a number of congruences and identities involving $NT(r,m,n)$. In this paper, we strengthen and extend a recent work of Mao on the transformation properties of the $NT$ function and provide an analogy of Hickerson and Mortenson's work on the rank function. As an application, we demonstrate how one can deduce from our results many identities involving $NT(r,m,n)$ and another crank-analog statistic $M_\omega(r,m,n)$. As a related result, some new properties of generalized Appell-Lerch series are given.