What is the ruling regarding someone who made a vow then forgot it, and predominantly believes it was a vow to fast, knowing that his intention was not firm at the time of making the vow?

If your predominant thought is that you made a vow to fast, then you must fast. However, if your thought is weak or equally balanced, then you must offer expiation for a broken oath. Doubt concerning the intention does not affect the ruling, as long as you uttered the vow with that specific intention.