Exegesis:

kai idōn tēn pistin autōn ‘and when he saw their faith.’ Whether autōn also refers to the paralytic cannot be decided, but it is not impossible.

pistis ‘faith’; here and in 7.9, 50; 8.48; 17.19; 18.42 of faith or confidence in Jesus’ power to heal.

anthrōpe ‘man,’ here used without the note of reproach which is clearly present in 12.14; 22.58, 60; hence “friend” (An American Translation, Willibrord, Bible de Jérusalem).

apheōntai soi hai hamartiai sou ‘your sins have been forgiven you.’ The agent of the forgiving is not mentioned, probably in order to leave room for the subsequent discussion with the Pharisees. Jesus never says “I forgive your sins”, and in v. 24 he speaks of his exousia, i.e. his authority derived from God to forgive sins (cf. on 4.32) and in the last analysis it is God who forgives the paralytic’s sins. For hamartiai cf. on 1.77.

Translation:

When he saw their faith, or ‘when he/Jesus saw this proof of their faith, or, realized/perceived that they believed in him.’

Man has been rendered e.g. by “(my) friend” (An American Translation, Phillips), ‘brother,’ a polite form of address (Bahasa Indonesia KB), ‘my son’ (Batak Toba), ‘little-one’ (Balinese, used by a father to his son, by a teacher to his pupil), an affective particle (Shipibo-Conibo). In some receptor languages the word of address simply has to be omitted (cf. on “Theophilus” in 1.3), or it can only be represented by a particle or form used to call attention to what the speaker is about to say (cf. Santali).

Your sins are forgiven you. The clause is a difficult one to render in languages without passive verbal forms, since a shift to an active with ‘God’ as agent is usually unacceptable because of v. 21. In such cases one should preferably use a rendering that does not explicitly mention the agent, e.g. ‘to receive forgiveness’ (Sranan Tongo), ‘forgiveness … has become’ (Marathi). Where it is obligatory to mention the agent one may say something like, ‘I declare to you, “God has forgiven your sins” ’ (which will require a corresponding adjustment of v. 21 (c)).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.