Among the others that bear the same epithet, Luke the Stylite is hardly the most well known. There is actually no reference for him either in BHO or in the BHG (an older edition) available to me as I write this. He is, however, celebrated on Kihak 17 (Dec 13), the day of his translation a couple of days following his death, in the Arabic synaxarion published by Basset (PO 3: 474-475), and I give an English translation of it here.
On this day we celebrate the translation of the body of St. Luke the Stylite. He was from the country of Persia. Then he was enlisted and became a commander [ʔamīr] over one hundred troops. After that, he left his command and everything else and strove to become a monk [qaṣada ‘l-rahbana]. He dwelt in a certain monastery of the east, where he stayed for some time. Then, when he had succeeded in becoming a monk, thanks to his merit, he was made a priest over the monastery. At the time of his dedication, he wore a suit of iron the size of his “seat” [b-qadri qaʕdatihi]. He persisted in fasting from that day, and he would fast for six consecutive days and break his fast on the seventh with a small piece of eucharistic bread [qurbāna ṣaġīra] and some green vegetables after having celebrated the eucharist and offering [baʕda quddāsihi wa-qurbānihi]. He ascended a pillar, on which he stayed for three years. Then he heard the voice of an angel calling him by his name and commanding him to come down. [The angel] showed him a cross of fire, so he went down and followed the voice, with the cross going before him until he reached a certain mountain, where he remained for a time. The people would come to see him and profit from his teaching. After that, he persisted in silence and put stones in his mouth, so that he could not speak to anyone. Then God revealed to him that he should go near Constantinople, so he went to a village nearby it. He ascended a pillar, remained on it for forty-five years, and strove in spiritual warfare [ǧāhada ǧihādan rūḥāniyyan]. God gave him favorable prophecy and the gift of miracles, and he would cure all the sick who came to see him. When God willed his rest from the toils of this world, he went to rest on the fifteenth of the month of Kīhak, and the one who had been attending him went and informed the patriarch and the clergy of his departure, so the patriarch took the clergy, the crosses, and the censers, and they went to his place, prayed over him, and carried him to Constantinople two days later, which was the seventeenth of Kīhak. Then they put him in the church [al-haykal] and concluded the prayer over him two days later, and the believers were blessed in him. Then they put him in a basin [? ǧurn] beneath the bodies of the saints, and from his body, God showed signs, miracles, benefits, and healings to everyone who came to see him in faith. May his prayers be with us!