A purely analytical extension of flattened Gaussian beams [Opt. Commun.107, 335 (1994)OPCOB80030-401810.1016/0030-4018(94)90342-5] to any values of beam order is here proposed. Due to it, the paraxial propagation problem of axially symmetric, coherent flat-top beams through arbitrary ABCD optical systems can definitely be solved in closed form via a particular bivariate confluent hypergeometric function.