In the letter, Jackson said his health makes it impossible to serve. He says he is doing his best to cooperate with a federal investigation of his activities.
The chamber was adjourned last week for the Thanksgiving holiday when Jackson resigned in a letter to Speaker John Boehner.
Meanwhile, Jackson's brother Jonathan indicates he is thinking about running for the 2nd District seat that is now vacant.
Jonathan Jackson told listeners on WVON radio Tuesday he doesn't know if he will run but will make a decision soon.
Democratic State Senator Toi Hutchinson is the latest to announce that she will run for Jackson's seat. That makes three announced candidates with several others on the fence.
A special election will be held early next year to choose a replacement.
The Associated Press contributed to this report.