Balthasar, Melchior and Gaspar?

Matthew 2:2 says: Some wise men came to Jerusalem from the east. Why are they traditionally said to be three kings, with the names of Balthasar, Melchior and Gaspar?

The Gospel tells us nothing about these men, and even the apocryphal writings do not add much. We have to speculate about who they were and where in the east they came from.

Wise men is a translation from the Greek word magoi. The term indicates men of a priestly caste common among the ancient Medes and Persians. The 4th century Roman historian Ammianus Marcellinus, who had served in the army in Persia, wrote that the Persian magi based their wisdom on the Brahmins, an Indian priestly order who were the most superior of the Hindu castes. Also, the 17th century Portuguese poet Luis de Camoes identified the magi as Indian Brahmins.

In the eastern Church there are traditionally 12 wise men. Our western tradition is that there were three, probably because of the three gifts they presented to Christ of gold, frankincense and myrrh.

Most of the other legends, such as their names, seem to have derived from the eastern Church, where it was customary to interpret scriptural events in a more symbolic way than in the west. Gold was regarded as a gift fit for a king, frankincense for God and myrrh as symbolic of Christ’s bitter death.
The names probably come from the tradition of the ancient king of Arabia (Bithisarea or Balthasar), king of Persia (Melichior or Melchior) and king of India (Gathaspa, Gaspar or Casper).

In the third century, Tertullian referred to the wise men as the Kings of the Magi, perhaps seeing a connection between them and the words of Psalm 72:10-11: The kings of Tarshish and of the islands will pay him tribute. The kings of Sheba and Seba will offer gifts; all kings will do him homage, all nations become his servants.