  A purely analytical extension of the flattened Gaussian beams [Opt. Commun.
\textbf{107,} 335 (1994)] to any values of the 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.
