At face value, The Gospel According to Luke purports to be a long letter to a friend of the author, Theophilus, who is presumably unsure (or indecisive) about the details in Christianity's foundational story. The author aims to set the record straight for Theophilus once and for all. Viewed in its epistolary function, this gospel is unlike the others in that it is explicitly addressed to a single individual in this way, and in that it also purports to describe the author's methods and purpose. The author of this gospel presents himself as an anthologist, an historian. This is no myth that he's relating; these things he describes are actual historical happenings, verified by "eyewitnesses," he says. One wonders how many people were questioning the veracity of the accounts in Mark and "Q"[4*] at the time that Luke wrote. Apparently, there were enough to inspire him to write his "definitive" version, lest Theophilus fall prey to these detractors' doubts.
With this apologetic aim in mind, Luke explicitly claims to be using eyewitness material in order that Theophilus, “may know the truth concerning these things.” He decided to write "an orderly account," implying that the versions of the story written before his own were somehow deficient or disarrayed. He doesn't specify what texts were in front of him as he wrote, but Mark and Q are surely among these sources he feels are less-than-adequate, in need of revision, of 'tidying up.'
What do we know about this author? An examination of internal evidence reveals a few things: First, his Greek is excellent, even elegant. Of all the gospels, in fact, Luke has the richest vocabulary.  He is versed in political terminology and uses it accurately. He is obviously well educated and very familiar with the writing norms and conventions of his time.
In weaving his tale, whereas Matthew follows the form of Mark closely, Luke strays from this order, favoring instead that of the hypothetical Q in the flow of his narrative. Some of the central organizational themes in Luke include: a concern with outcasts; the “Spirit” as a "character" in the gospel; the centrality of Jerusalem, and, ironically, the simultaneous centrality of gentiles as co-inheritors in God’s salvation scheme. While he does seem to have some knowledge of Judean ideas, especially regarding the soteriological messianism mentioned previously, his repeated insistent championing of the gentiles as co-inheritors of God's salvation makes it very likely that he is of the gentile variety of Christian. Even when some apologists attribute the other three gospels to Judean writers, as is the norm, that of Luke is often recognized as being the work of a Greek for this reason.
Where does Judaism factor in?
Luke’s genealogy, instead of going all the way back to Abraham as Matthew‘s does, goes all the way back to Adam, that is, to humanity‘s very beginning. Whereas Matthew’s birth narrative is a midrash on the Story of Moses, the birth narrative in Luke is less regal, less extravagant, more down to earth and humble. An angel announces to shepherds (not to kings or exotic Persian astrologers, as in Matthew) that the savior has come. For Luke Jesus is an Elijah-type all-inclusive savior who will save the world—the whole world, gentiles included. This stress on radical inclusivism is there from the beginning, always supporting the notion that Jesus has come not just for Jews but for all people. Old man Simeon (a Jew) recognized Jesus as “a light for revelation to the Gentiles”when Mary and Joseph had brought him to the Temple to be circumcised.
Had there been a significant Jewish population in the community that Luke represented, I doubt that he would have stressed the gentile role in salvation as much as he does, especially at a time (consensus says 85–90 CE) when Christians were reportedly being kicked out of the synagogues and even ritually cursed.
Typology is as important a hermeneutic tool to Luke as it is to Matthew. But in Luke’s case the chosen typology is that of Jesus as the new Elijah, the super prophet. That character also appears and figures in the stories told in Mark and Matthew, but in those two gospels, the Elijah/Elisha cycle type is sometimes applied more to John the Baptizer, as a kind of herald of Jesus. In Luke the type is always imposed on Jesus himself.
Some of the titles that the author of Luke applies to Jesus reflect some of the other Hellenistic elements he associates with him. For example, of the synoptic gospels, only Luke’s applies the title soter (savior) to Jesus. This was a common designation on many of the surviving inscriptions of ancient Rome. As part of the honor/benefactor culture of the Mediterranean region, the title would be bestowed on benefactors and sponsors. In its first century context, the word was often used literally. A savior would “save” one, from disease, from foreign invasion, from hardship, from slavery, etc. This was certainly not a new concept in the Greco-Roman world. It’s a down-to-earth salvation the word implies in the Greco Roman context, not the sophisticated, convoluted other-worldly salvation that the word implies in the Christianity we grew up with. The Jews had these savior types as well (Moses, Joshua). The title was in fact conferred on a bunch of ancient figures. Gods and leaders especially would receive it often (Pompey, Asclepius). Augustus, in fact, was regarded as a savior and as “son of god.” What’s more, the period of peace and prosperity that he was credited with ushering in was regarded by many as “evangelion,” that is, as “good news.” It may come as a surprise to many Christians, but these titles and words were demonstrably inherited from a Hellenic matrix, not a Judean one..
The first time that Jesus speaks publicly after his baptism, he is given the scroll of Isaiah at the synagogue in Nazareth.
"The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, to preach the acceptable year of the Lord.
And he closed the book, and he gave it again to the minister, and sat down. And the eyes of all them that were in the synagogue were fastened on him.
And he began to say unto them, This day is this scripture fulfilled in your ears. And all bare him witness, and wondered at the gracious words which proceeded out of his mouth. And they said, Is not this Joseph's son? " (Luke 4:18–22)
In his interpretation, Jesus seems to be staking a claim for this exalted position as the abstract savior messiah that I profiled previously, the anointed one of God. Everyone is impressed by his announcement and they start to wonder at his authority. “No prophet is accepted in his home town,” he says. He thus refers to himself as a prophet in Luke, and immediately mentions Elijah (4:25).
One of the texts relevant to the Elijah typology is in Malachi:
"Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD — (Malachi 4:5)"
Luke stresses the Elijah connection in another way by paralleling a story from 1st Kings in which Elijah visits a widow, whose son has died. Elijah raises the son from the dead. Likewise, Jesus revives a dead man upon entering the town of Nain with his disciples (7:11–16). Fear seized all those who witnessed it and they glorified God, saying, “A great prophet has arisen among us, and god has looked favorably on his people.”
Luke is a highly educated gentile Christian writing to a gentile Christian conscript. The central theme in Luke is the legitimization and inclusion of the gentiles. He keeps harping on it. Once again, we run into the puzzling phenomenon of someone using Jewish symbols and citations to demonstrate to gentiles that Judaism has been superseded by radical universalism (shhh . . . don't tell the rabbis!).
There is no mention of James in the Gospel of Luke, despite the gospel's emphasis on the centrality of Jerusalem in the ministry and death and resurrection of Jesus. This is not as problematic as it is in Matthew, but it is still cause for pause.
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1 - Luke uses 2055 words, of which 957 (nearly half!) are hapax legomena.— i.e. a fancy way to say that a word only appears once in a text. This is over 800 more than Matthew.
2 - Luke 2:21–38 — Simeon recognizes Jesus, who has been brought to the Temple to get circumcised. Ironically, he here proclaims Jesus the savior of the uncircumcised gentiles.
3 - The addition of the "benediction against the minim," included in the Amidah, a Liturgical cycle, as the twelfth benediction. This prayer dates to around this period that consensus places the composition of this gospel, as does the hypothetical "Council of Jamnia," which was conjectured to explain this overt expulsion and open condemnation of the Christians that occurred at the time. I will have more to say about this in a future post.
4 - Addendum: If Mark Goodacre and Burton Mack are correct about Q being an unnecessary theory, all that changes here is that Luke's deviation from Mark was his own invention. They may be right.