Are the oath of persistency (yamin al-lajaj) and the vow of persistency (nadhr al-lajaj) the same, and is expiation obligatory for repeated oaths of persistency concerning the same sin, with a mention of the evidence that it is a vow?

A vow of insistence (Nathr al-Lajaj) is when a person prevents himself from something or urges himself to do something by making a commitment to perform an act of worship (qurbah), such as saying: "If I do such-and-such, then I owe Allah such-and-such." This is also called an oath of insistence (Yamin al-Lajaj). Its ruling is that one has the choice between fulfilling the vow or offering expiation for an oath (kaffarat yamin), because it resembles a vow in terms of committing to an act of worship, and an oath in terms of prevention. An example of this is someone who says: "I owe Allah a vow that if I do such-and-such, I will fast for three days," and then he does that thing. He then has the choice to fast for three days or to offer expiation for an oath. This applies to your oath, so you have the choice between fasting for three days or offering expiation for an oath. Some scholars hold the view that fasting for three days or one expiation for an oath is sufficient for all the oaths you have broken.