While it is clear that παις is one of the two most frequent categories in this unit it should not be simply equated with Deutero-Isaiah’s ‘ebed. . . . one factor immediately erodes so simple an equation—David is unambiguously called παις at Acts 4.25 and two of the references to Jesus as παις follow it immediately in the same context—that of worship. There is a strong likelihood that what is meant by παις in one Lukan use is what should be read in the other. So either David is to be thought of as a servant figure, which seems unlikely, or the word refers more generally to a man who in some way is God’s man. It may have the overtone of ‘son’ here, in which case, again, it is not the ‘ebed Yahweh modern which is at work.
The strongest argument for παις at Acts 3.13 to be read as ‘ebed Yahweh is probably its association with Iδοὺ συνήσει ὁ παῖς μου καὶ ὑψωθήσεται καὶ δοξασθήσεται σφόδρα (Isa. 52.13). . . . The presence of εδοξασεν may or may not, be confirmation that Luke had the ‘ebed Yahweh in mind here: certainty is not possible. But the references to δοξα may be much wider than this one Isaianic reference and relate more to what was known of Jesus’ life—including the transfiguration narrative—than to the ‘ebed Yahweh. At present a connection with Isaiah is simply not proven; but another factor needs to be noted. In the little doxological passage (4.23-41) which speaks of David as παις, Jesus is called:
--τον αγοιν παιδα σου ‘Ιησουν (4.27)
--του αγιου παιδος σου ‘Ιησου (4.30)
Nobody, it seems, argues as they do for δικαιος, that αγιος is a function of παις in Deutero-Isaiah; but this christological formula, set in the context of worship, cannot be divorced from the clear reference to David at Acts 4.25.
A final factor needs to be taken into account: even if παις at Acts 3.13 should, on balance, be taken to be an allusion to ‘ebed Yahweh, there is no necessary connection with δικαιος in the following verse where it is linked with αγιος. Indeed . . the ‘ebed Yahweh is only once called δικαιος—that is, δικαιος is not a major qualifier in the model. There is, however, strong evidence from both Paul’s and Stephen’s speeches [that] Luke worked with a christological modern in which δικαιος was both primary and relatively frequent. (Peter Doble, The Paradox of Salvation: Luke’s Theology of the Cross [Society of New Testament Studies Monograph Series 87; Cambridge: Cambridge University Press, 1996], 153-54)