Need a forum with user integration

[comtwous]

Want to build a forum (name it `F`) for mobile web application so that:

* a registered user on my existing website (name it `W`) can log in onto `F` directly using the user ID and password pair registered in `W`, without registering again on `F`. Didn't come up with an example to proof the concept yet. But the business logic should be roughly like this: a user log in website `W` i, click on a `Go to Forum` button on his or her setting page, the `F` forum page get launched in mobile web browser, and the user get logged in the `F` forum automatically with information registered on `W` website, which can be user ID and password pair, or the cookie, if possible, and so on.

* mobile devices such iPhone and Android phones should load pages on `F` naturally. [MacRumors forum](http://forums.macrumors.com/) serves as a good example: we get mobile styles when opening the forum on mobile devices.

Have investigated into vBulletin 5 but seems like it's not so good at doing the above things. Has anyone implemented the above like forum using vBulletin 5?