The double appropriation of this aspect of the David-traditions happened quickly, within two-plus decades of Jesus’ death. Already by the late 50s of the first century, on the evidence of Paul’s letter to Rome, the association of Jesus with David, both past and future, was firm. Paul uses it as the springboard for his own self-introduction:   

Paul, slave of Jesus Christ, called to be his messenger, set apart for God’s good news— 2 promised beforehand through his prophets in the sacred scriptures— 3 the good news concerning his son, from the seed of David according to the flesh, 4 and declared son of God in power according to the spirit of holiness by the resurrection of the dead: Jesus Christ our lord, 5 through whom we have received grace and apostleship in order to bring the obedience of faithfulness on behalf of his name to all the nations/gentiles/pagans, 6 including to you. (Romans 1.1–6; my translation)   

Paul simply asserts Jesus’ Davidic lineage here. He does not argue it, but ties it immediately to Jesus’ impending apocalyptic reappearance, when the dead would be raised. The claim of Jesus’ past Davidic descent, in other words (and for all the interest that Paul shows in it here), seems “caused” by expectations about Jesus’ future eschatological performance. - Paula Fredricksen, When Christians Were Jews, p.102-103