This record is in Luke 3:23-38. Luke was a physician. He carefully investigated the life of Christ, and wrote the books of Luke and Acts (Colossians 4:14, Luke 1:1-4, Acts 1:1).
The original readers of Luke's works were Greek Christians. While Matthew wrote to the Jews, Luke wrote to the Greeks.
Matthew's genealogy emphasized Jesus' claim to the throne of David. Since Luke's readers were less concerned about the fulfillment of Jewish prophecy, his genealogy focused on Jesus' descent from God. It placed no emphasis on Jesus being the descendant of King David.
Reason for Differences
Differences between the genealogies of Matthew and Luke may be attributed to the fact that Matthew traced the ancestry of Joseph, while Luke traced the that of Mary.
Unique Placement of the Genealogy
The placement of Luke's genealogy is after the baptism of Christ. When Jesus was baptized, God said "This is my beloved son." Immediately following this event, as if to prove God's declaration, Luke inserted the genealogy.
Encouragement to Greek Christians
The genealogy culminated by showing Jesus was the "son of Adam, the son of God" (Luke 3:28). This emphasized the humanity of Jesus, and the equality of all christians, regardless of ethnic backgrounds.
Christians of Jewish descent originally considered Greek Christians as inferior to themselves (Acts 15:1-31, Galatians 2:11-16). Luke's genealogy underscored the fact that Jesus was the son of God. Since all men are made in the image of God (Genesis 1:26-27), this may have been a source of encouragement to the Greek Christians.