Is there a difference in meaning and use between these sentences?
- See you next week.
- I'll see you next week.
- I'll be seeing you next week.
When I say good-bye to a friend, I tend to say (1). I guess (1) is probably the short version of (2) and so they probably mean the same thing. (Please correct me if I am wrong). I have heard (3) as well, but I'm not sure whether it has a special meaning attached to it.
Yes, (1) is a short form of (2). The subject I and the auxiliary verb will are both understood to be elided. (Otherwise it wouldn't be grammatical: a future tense needs will and sentences need subjects.)
"I will be seeing you next week" is almost the same; however, there is slightly greater feeling of certainty. Your doctor might say that if you have a follow-up appointment in one week's time to emphasize that, or that same person could just use "see you next week".
By contrast, it would be somewhat unusual to say "I will be seeing you next week" to coworkers on Friday evening, whereas "see you next week" is fine. The reason is that we don't have a specific appointment to meet coworkers next week; it just happens because we show up for work. It's simply too many words to use for something that is passively expected to occur.
The last one is continuous which could mean that you will be seeing them more than once, that you seeing each other is going to continue for a while, but there is also aspect which can be chosen by the speaker to convey more eternal feeling to the action.
