Yes you can throw a football after a fumble as long as it is fumbled behind the line of scrimmage and it is the only forward pass of the play. Each offensive play is allowed only one forward pass.
A ball advanced beyond the line of scrimmage may be passed or fumbled backward behind the line and then legally thrown forward as long as the feet of the passer are behind the line when he throws the ball and it is the first forward pass during that play.
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On any down, if the offense recovers their own fumble, they can advance it as far as they can until tackled, or run out of bounds.