Our HOA sent notices that stated that any home owner who had Christmas lights up at the end of January would be subject to a weekly fine. If this is not specifically stated in the bylaws, is it legal to impose this fine?

A homeowner who has not removed his extgerior Christmas decorations by the end of January is deserving of flak from his HOA.
He has a responsibility to his fellow homeowners to abide by the associations covenants, bylaws, rules, etc., etc.
The HOA is legally able to impose fines and penalties on the homeowner accordingly (see preceding paragraph).
Why would you want to be part of a problem instead of being a responsible homeowner?

While it might not be specifically stated in your HOA bylaws, as in the exact words, it might be an interpretation of specific bylaw. For example, I lived in a unit where the HOA rules stated that all units must appear uniform on the outside. An interpretation of this general bylaw was that all window treatments must be white in appearance on the outside. So you could have fuschia curtains if you wanted, but they could not show any color on the backs. I'm not making this up. We all got notices stating we had 30 days to remove and replace any window treatments that did not meet that description or we would be fined.

Quite frankly, I cannot imagine that anybody would want to leave their Christmas lights up until past the end of January anyway. You might contact your HOA and ask them what the basis for this rule is and where it can be found in the bylaws. Explain that you were not able to find the specific passage that addressed Christmas lights.

Most HOAs are smart enough not to send notices that threaten fines without having the bylaw on paper. If it isn't written into the bylaws directly or indirectly, then the fines can be challenged.

Laws and bylaws are different everywhere, so check with your HOA and see if they can explain their notice and cite where it is addressed in the bylaws and take it from there.