I'm not sure where you've gotten your information (i.e. to green) about the purpose of end season fertilization, but fall and winter applications should serve the purpose of strengthening the plant and building food stores to, repair summer damage, reduce stress during the Winter, and prepare for early Spring green-up and root growth.
Some experts recommend a 3-1-2. I have trouble finding that ratio and drop a triple slow release N as my first and second early (Sept.) and mid fall (Oct.) app. (T10 or T12) The phosphorus benefits photosynthesis and cell division and the potasioum benefits carbohydrate production. My third drop, Late Oct. early Nov. in N.E.Ohio is a fast release N 30-0-0 or 30-0-5. Fast release to make the N available for quick turf grass uptake for carbohydrate storage. I apply at 1/2# per K ro reduce incidence of snow mold that has been attributed to late season N apps.