Lexical Definition

‎ἀνθομολογέομαι‎ anthomologéomai (anth-om-ol-og-eh'-om-ahee) to praise ^G:V
from ‎ἀντί‎ and the middle voice of ‎ὁμολογέω‎

give thanks.

• to confess in turn, i.e. respond in praise

Abbott-Smith

ἀνθ-ομολογέομαι, -οῦμαι (ἀντί, ὁμολογέομαι), [in LXX: Ps 78 (79):13* (יָדָה), Da LXX 4:34* (שְׁבַּח), I Es 8:91, Si 20:2, III Mac 6:33*;]
1. to make a mutual agreement (Dem., Polyb.).
2. to acknowledge fully, confess (Diod., Polyb., cf. I Es 8:91).
3. C. dat. pers., to declare one's praises, speak fully in prayer or thanksgiving, give thanks to (cf. Ps 78 (79):13*): Lk 2:38 (Cremer, 771; MM, VGT, s.v.).†

Abbott-Smith References

Liddell-Scott-Jones

ἀνθομολογέομαι-έομαι,

make a mutual agreement or covenant, πρός τινα[Refs 4th c.BC+]

__II confess freely and openly, τὰς ἀρετάς τινος[Refs 2nd c.BC+]
__II.2 admit, signify, πρός τινα μηδὲν ἑωρακέναι[Refs 1st c.AD+]
__II.3 assent, agree, τοῖς εἰρημένοις[Refs 2nd c.BC+]
__II.4 return thanks to God, [LXX+NT]; χάριν ἀ. return thanks, [Refs 1st c.AD+] —active, -λογέω admit a claim, is late, [Refs 3rd c.AD+]

_{LSJ Scripture Refs: LXX.Ps.78.13* ☩ Luke.2.38}