Man, they have let go of themselves for you, those mistakes of yours,
It is interesting to compare this verse to the related versions in Matthew (Matthew 9:2) and Mark (Mark 2:5). The differences could mean Christ said something like this three different times of that these are all variations on a theme, leaving us to guess which is the most accurate. Mark is the shortest. Both Matthew and Luke here have words not in the others. The Matthew version is longer, having a phrase in the beginning that is missing from the other two. This might indicate that the others are shortened versions of the original in Matthew.
The Greek word for "man" also means "person" and "humanity" in the singular. In other versions of this verse, the word used is the Greek word for "child".
The "thy" here is the common Greek pronoun. In the source here, it appears after "sins", which is a common pattern for Christ. In Matthew and Mark, it appears before the word.
The word translated as "sin" is a form of a word that means "to fail in one's purpose", "to neglect," and "to be deprived of." It has no sense of doing malicious evil in Greek. The best English translation is "mistakes" or "failures" rather than what we commonly think of as the evils of "sin." See this article for more information and context.
The word translated as "are forgiven" primarily means "to let go" or "to send away." This same word is usually translated as "leave", "forgive", "suffer," and "let" in the New Testament. The form here is a past tense, where as in the other two versions, it is present. As with the other versions, the form isn't passive, but a form where the subject acts on themselves. "Have let go of themselves".
The "thou" here is the second person pronoun in the form of an indirect object, which in Greek can be used to mean "for you" or "by you" as well as "to you." This down
An interesting contrast of three different versions of this verse.
ἀφέωνταί (verb 3rd pl perf ind mp) "I leave" is aphiemi, which means "to let fall", "to send away", "give up", "hand over", "to let loose", "to get rid of", "to leave alone", "to pass by", "to permit," and "to send forth from oneself."