Help with a sitch....
 

Follow along closely:

A1 has the dribble on the wing outside the three point line. A1 dribbles in to the elbow area, and loses his footing trying to dribble around defender B1. A1 falls to the floor, losing control of the ball completely, then gets back up, regains possession of the ball, and dribbles in for the layup. No other player touched the ball during this sequence.

Whatcha got?

Depends on what you mean by "regains possession".. Did he pick the ball up or just bat the ball to the ground, continuing his dribble? If he picked it up then his dribble ended (see NFHS 4.15) and if he dribbled again you have a double dribble...if he just batted the ball to the floor, continuing the dribble then you have nothing, just an interrupted dribble that was continued..

How did A1 regain possession?

This is an interrupted dribble, so A1 may continue dribbling as long as nothing is done that would end the dribble (such as picking it up with both hands).

I agree. You'd have to see how possession was regained to make an accurate call on this one. Did the player grab the ball or begin to dribble again?

An interrupted dribble. If he continued the dribble it would be legal. If he grabbed the ball with both hands first, then dribbled again it would be double dribble.