The Fourier-Jacobi expansion of the singular theta lift

Recently, Funke and Hofmann constructed a singular theta lift of Borcherds type for the dual reductive pair \(U(1,1)\times U(p,q)\), \(p,q\geq 1\), the input functions of which are harmonic weak Maass forms of weight \(k= 2-p-q\). In the present paper, we give an explicit evaluation of the Fourier-Jacobi expansion of the lift. For this purpose, we adapt a method introduced by Kudla in his paper 'Another product for a Borcherds form'. As an application, in the case \(U(p,1)\) we recover a new infinite product expansion associated to a Borcherds form, analogous to the case \(O(p,2)\) treated by Kudla.