This passage occurs as a New Testament Lesson in the RCL "Pentecost"/"Ordinary"/"Proper" Season, Year A, most recently October 2014.
Summary: As I have stated previously, I view Paul's letter to the Philippians as a small treatise on sanctification. You can find beautiful fruit in these passages, beloved words that evidence the Spirit's work in Paul to make him a little Christ for all of us. What struck me this time around though was the profound way in which the community in Christ takes precedence in this passage. Paul continues to offer many images of working together and community love, even calling his fellow Christian his "desired." Paul doesn't conclude with his love of Christ, but the love Christ has given him for his fellow believers. Our sanctification is precisely this: OUR sanctification as the Holy Spirit moves us closer together in love and hope.
αγαπητοι and επιποθητοι ("beloved" and "desired" 4:1) αγαπητοι is probably familiar enough to most Christians, especially those who work with Greek. Paul calls his brothers and sisters in Christ who beloved. Wow! Yet, επιποθητοι is more startling. This word comes from desire. While we have seen the root verb elsewhere in Philippians (1:8; 2:26), no where else in the Bible do we find this term επιποθητοι! This sense of desire can be positive, for example, the deer pants for the water like the soul desires God (see Psalm 84:3/Psalm 41:2). However, Paul here is claiming the other Christians are his desired. This truly is taking the mind of Christ -- when we love each so deeply that we can talk about a deep love for one another. What does the mind of Christ and sanctification mean? It means loving your neighbor, so much, that you desire to be with them like Christ desires to be with them, like their soul desires to be reunited with God.
και σε ("even you", 4:3) Paul generally speaks in the second person plural throughout the letter. Perhaps he is writing to someone specific; maybe he wants to drive home that these words are for each person. Maybe its ambiguous so we all think, well, its my job to help those two women who are fighting.
συζυγε; συνηθλσαν; συνεργων ("yoked", "co-striving" and "co-worker", 4:3; the second is a verb, the other two adjectives) Paul here presents us with a few images of the Christian life. The first is from the idea of a yoke and can actually refer even to marriage. The image of oxen plowing the field. The next is to atheletes in contest with one another. The last is co-worker, perhaps the least descriptive, but you put the three of them together and Paul profoundly gives us some images of our life together!
γνωσθητω ("let it be"; imperative (command), 4:5 and 4:6). There first time Paul uses this verb, it is telling us to let our gentleness be known to all people; the second time it is Paul telling us to let our prayers be known to God. In this context though, I wonder if they are so exclusive. I wonder if we read this through a western-post-enlightenment idea of worship that would have our prayers of thanks be those in private. Part of our joy and duty, as Psalm 66 suggests, is not simply praising God in private but offering thanks in front of the congregation.
Grammatical review: "αυτο"
The word αυτο and its various conjugated forms (αυτου for example) can be a bit tricky for the reader. First because another set of words, meaning this and that, looks very similiar but have different accents. But it is also tricky because the word αυτο can mean three different things, even if it looks the same.
It can function like a pronoun: αυτου for example, almost always means "him." In this case, the pronoun is in the genitive, so it fully means "of him." It functions this way 95% of the time.
It can also mean "very." This is when it stands alone (predicate position). This is fairly rare. An example of this is in Philippians 1:6 πεποιθως αυτο τουτο: "I am convinced of this very thing."
It can also mean "same." It behaves like this when it follows an article. Hence, in Philippians 4:2 you get: το αυτο φρονειν: "The same thinking." Paul actually uses this also in 2:2 and 2:18.
Again, for 90-95% of translation, the word functions as a pronoun, but it can be helpful to remember these other uses.