What is the ruling regarding children who were born during a period when their father had divorced their mother more than three times and then taken her back, and are these children considered legitimate, and what are the Shariah implications for the parents?

The prohibition of swearing by other than Allah is well-established, and this prohibition is even stronger regarding oaths of divorce. If the father breaks three oaths of divorce, intending thereby to effect a divorce, then the divorce occurs, and the mother becomes irrevocably divorced from him (baynunah kubra). She is not permissible for him until she marries another man in a valid marriage, consummates it with him, and then he separates from her. As for if he swore intending to encourage or prevent something, and not intending the divorce to occur, the ruling is the same according to the majority of scholars. It is obligatory for the father to separate from the mother because she is a stranger to him, and if intercourse occurs between them, they must repent and seek forgiveness, because it is consideredzina (fornication/adultery). Sheikh al-Islam Ibn Taymiyyah held the view that whoever swears by divorce intending to threaten or prevent something, and then breaks the oath, is only required to offer an expiation for an oath (kaffarat yamin). As for the children conceived from his intercourse after he divorced her three times, they are attributed to him. In this case, he would be subject to a legal punishment (hadd) if he was aware of the prohibition of what he did.