How do I ask my professor to recommend me to a few specific industry positions?
It is appropriate for you to ask for a recommendation. However, the professor may not feel it is appropriate for him to ask them directly. But it is part of their responsibility to at least write you a letter of recommendation. And they may be willing to do more.
But the &quot;way&quot; to ask, is just to ask. Ask for a recommendation, preferably a letter and something more personal. It is up to them how to proceed after that.
I'd guess that even a letter would be pretty effective given the relationships.