Luke 13:25 ἀφ᾿ οὗ ἂν ἐγερθῇ ὁ οἰκοδεσπότης καὶ ἀποκλείσῃ τὴν θύραν καὶ ἄρξησθε ἔξω ἑστάναι καὶ κρούειν τὴν θύραν λέγοντες· κύριε, ἄνοιξον ἡμῖν, καὶ ἀποκριθεὶς ἐρεῖ ὑμῖν· οὐκ οἶδα ὑμᾶς πόθεν ἐστέ.
I habitually compare Grimm-Thayer to Danker'r 3rd Ed. and in case Thayer seemed to be saying something different than Danker. I don't know what to make of Thayer's comments:
ἄρχομαι is connected with an infinitive and that so often, especially in the historical books, that formerly most interpreters thought it constituted a periphrasis for the finite form of the verb standing in the infinitive, as ἤρξατο κηρύσσειν for ἐκήρυξε. But through the influence principally of Fritzsche (on Matthew, p. 539f), cf. Winers Grammar, § 65 7 d., it is now conceded that the theory of a periphrasis of this kind was a rash assumption, and that there is scarcely an example which cannot be reduced to one of the following classes:
a. the idea of beginning has more or less weight or importance, so that it is brought out by a separate word: Matthew 11:7 (the disciples of John having retired, Christ began to speak concerning John, which he did not do while they were present); Luke 3:8 (do not even begin to say; make not even an attempt to excuse yourselves); Luke 15:14 (the beginning of want followed hard upon the squandering of his goods); Luke 21:28; 2 Corinthians 3:1; especially when the beginning of an action is contrasted with its continuance or its repetition, Mark 6:7; Mark 8:31 (cf. Mark 9:31; Mark 10:33f); or with the end of it, Luke 14:30 (opposed to ἐκτελέσαι); John 13:5 (cf. 12).
b. ἄρχειν denotes something as begun by someone, others following: Acts 27:35f (Winers Grammar, § 65, 7 d.).
c. ἄρχειν indicates that a thing was but just begun when it was interrupted by something else: Matthew 12:1 (they had begun to pluck ears of grain, but they were prevented from continuing by the interference of the Pharisees); Matthew 26:22 (Jesus answered before all had finished), Matthew 26:74; Mark 2:23; Mark 4:1 (he had scarcely begun to teach, when a multitude gathered unto him); Mark 6:2; Mark 10:41; Luke 5:21; Luke 12:45; Luke 13:25; Acts 11:15 (cf. Acts 10:44); , and often.
d. the action itself, instead of its beginning, might indeed have been mentioned; but in order that the more attention may be given to occurrences which seem to the writer to be of special importance, their initial stage, their beginning, is expressly pointed out: Mark 14:65; Luke 14:18; Acts 2:4, etc.
e. ἄρχω occurs in a sentence which has grown out of the blending of two statements: Matthew 4:17; Matthew 16:21 (from ἀπό τότε ἐκήρυξε ... ἔδειξέ, and τότε ἤρξατο κηρύσσειν ... δεικνύειν). The infinitive is lacking when discoverable from the context: ἀρχόμενος, namely, to discharge the Messianic office, Luke 3:23 (Winer's Grammar, 349 (328)); ἀρξάμενος namely, λέγειν, Acts 11:4. (Compare: ἐνάρχω (ἐνάρχομαι), προενάρχομαι, ὑπάρχω, προϋπάρχω.