There are at least two reasons why you might want to consider growing your tuberous begonias over the winter under fluorescents. Take cuttings of favourite plants in the fall just before I start the process of putting the begonias to bed for the winter. The cuttings will provide you with additional plants identical to the parents at a very low cost. The second option when receiving tubers in the fall is to place them in a vegetable crisper and store them in the fridge until the following spring. Assuming the tubers don't freeze, they should be ok.

However, you can artificially create a good enough growing environment allowing you to grow them right through the winter. Not only does this provide you some nice flowers by January, but also give you the opportunity to further increase your supply of plants with yet more cuttings. Not a bad deal for the small price of the hydro it takes to run the fluorescents. 

Light and Temperature Issues
Begonias are phototropic, in other words they do different things depending on how much light they get. Use a double bank of 40 watt fluorescent light tubes and leave them on between 16 - 20 hours a day. The dark time gives the plants some time for the roots to respire. As bottom leaves don't get near the amount of light as the ones near the top do, anything you can do in the way of providing additional ambient side lighting will be appreciated by the plants. Also, keep the temperatures above 60 degrees F if you can. Much below that it seems to noticeably slow down active growth.

Winter Tuber Maintenance
On larger tubers where you can identify extra "eyes" near the base of the basal shoots, take cuttings off the first set of shoots and the tubers will invariably throw up another shoot in response. However, at this point you'll want to make the decision of whether to grow your fall cuttings and tubers for simple tuber maintenance or for flowers. If you're focusing on propagation cuttings, you should go with a tuber maintenance strategy for the remainder of the winter. Specifically:
1. Pinch all flower buds.
2. Allow the plant to grow only to about about 10" high, then pinch the growth tip to keep it there.
3. For food, use something with a high potassium content.

For Flowers
By doing everything right, your plants will still not be nearly as nice or the flowers nearly as big as when you grow them naturally during the summer months. The brightness and colour temperature of the fluorescents is not quite there, air movement and ambient room temperatures are probably not what the should be either. But knowing these issues in advance allows you some leeway, specifically:
1. On each plant, pinch all flower buds except one - only allow one flower per plant.
2. Once the flower is spent, cut off the growth tip of the plant and switch to a high potassium fertilizer.

These few tricks will ensure you don't overstress the plant and allow you the extra time to fatten up the tuber for it's true spring planting.