Electron-quasihole duality and second-order differential equation for Read-Rezayi and Jack wave functions

We consider the quasihole wave functions of the non-Abelian Read-Rezayi quantum-Hall states which are given by the conformal blocks of the minimal model WA_k-1 (k+1,k+2) of the WA_k-1 algebra. By studying the degenerate representations of this conformal field theories, we derive a second-order differential equation satisfied by a general many-quasihole wave function. We find a duality between the differential equations fixing the electron and quasihole wave functions. They both satisfy the Laplace-Beltrami equation. We use this equation to obtain an analytic expression for the generic wave function with one excess flux. These results also apply to the more general models WA_k-1 (k+1,k+r) corresponding to the recently introduced Jack states.