Fourier coefficients of Jacobi forms of real weights

Let m and N be positive integers and k be a negative real number. Let [Formula: see text] be a multiplier system of weight k on [Formula: see text]. In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight k, index m, and multiplier system [Formula: see text] on [Formula: see text]. As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity.