How to approach undergraduate research without official program?
Go and talk to the relevant professors. 

It itself, seeking to make contact is hardly disrespectful -- so long as you are polite of course! Nor is it arrogant -- unless you show up and start teaching the professor their job. And as a university student, you should feel some entitlement to engage with the faculty beyond normal office hours or lectures -- so long as you don't abuse that access.

If you are concerned about appearing brash, send a brief, polite email asking for a meeting to discuss the mathematics you are interested in. If your professors do not want to set aside time to chat with a keen and driven undergraduate student about topics of mutual interest, then more fool them. Bear in mind, always, that professors often have a great deal of pressures on their time. However, I'm willing to bet that chatting about research would be a far more pleasurable activity than many other tasks a professor would have to tackle in the course of his or her day.

Personally, I make time to chat with any undergraduate student about the research I do, and any research they have done on their own accord, or how to further their interests.