This sounds like an interesting problem. Based on your description, this may be no big deal, or, you could have a real underlying problem that is showing its face as purple fingertips. Thus, you will need to have this evaluated by a general practitioner first before seeking an opinion from a specialist.
You could have a very mild case of Raynaud's phenomenon. This is where the blood vessels that bring blood to the tips of the fingers contract more than they are supposed to in response to cold weather. In its most severe form Raynaud's can cut off blood flow to the point where tissue in the fingers begins to die. You did not report any numbness or tingling in this area which suggests that the blood flow is still adequate. I don't think this is related to your arthritis
I suggest that you schedule an appointment with your primary care physician
. He or she can examine your finger tips and determine whether this could be Raynaud's phenomenon. If this is the case, then you should make extra efforts to keep your hands warm in the winter. Your doctor
will also likely order some blood tests to look for some of the causes of this problem. Good luck.