We don't even have a definition of "productivity" in the realm of programming, so I don't know what anyone would study. I have made a good living on a laptop for years, with no external monitor or any other accessories. I feel productive with this setup, and when I had multiple monitors years ago I didn't feel any boost. I noticed many of the other programmers keeping Facebook or Twitter open on their second monitor at that job, which may have caused a productivity hit, I don't know, because at that job, like every job I've had in my career, we didn't have any way to measure productivity.
It's been a while but there definitely are studies. For most "knowledge workers" the biggest jump in productivity was going from one to two; for programmers having three was a non-trivial productivity boost. All of that said, there are times I unplug the laptop from the dock so I can have just one app -- full screen -- in order to concentrate without distractions. Depends on the task but in general two monitors is the sweet spot.
I like one large monitor where windows snap to quarter/half screen