Politely ask a professor about my application status
I think that's too formal, you want to use a more informal communications channel such as email and just say "just wanted to ask if you know the status of my application" period.   That gives him the option of gracefully declining to answer.  You do not know what internal politics are going on in the program and what they are going to do with your application and there's really no way of forcing them to do anything.  All you really need to do is confirm that they got it.  Hopefully you are also aware that you should be looking at other programs and not setting your heart on just this one.

The other benefit of using an informal approach is that you can repeat it without appearing to be obnoxious.  Send him an email now, then keep doing it every couple weeks.  Sending a big old formal letter is kind of demanding a formal decision and response and puts them on the spot, and if you keep sending such letters then you come across as rigid and demanding.  Whereas an informal one-liner email isn't going to be perceived as demanding (it might be perceived as begging, but that's OK)