From the Super Mario Wiki, the Mario encyclopedia
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The Bill Gun is Roy Koopa's signature weapon in New Super Mario Bros. U, New Super Luigi U, New Super Mario Bros. U Deluxe, and Paper Mario: Color Splash. It is a longer, bazooka-like version of the Bill Blaster cannon, and when Roy fires a Bullet Bill from it, the force from it pushes him backwards a small distance and prevents him from moving, leaving him vulnerable to jumps. It replaces Roy's magic wand from the other Super Mario games.
When Roy is hit, he lets go of the gun but it still stays in his hand. The gun "travels" with him in the same direction as him in his defeat animation despite having his hands open and him not holding it.
In Paper Mario: Color Splash, Roy transforms his magic wand into a Bill Gun, which he uses to launch Mario's stolen paint at him. When the Washing Machine Thing Card is used after Roy darkens the room with black paint, the Bill Gun is destroyed.