The Northern Irishman officially took charge at Anfield last Friday deciding to leave South Wales after seventeen months.
After a successful spell at the Liberty Stadium he admitted the chance to manage Liverpool was too good to turn down.
"The chance to manage one of the world’s great footballing institutions doesn’t come around very often in a manager’s life and, after two approaches, I have decided to take up the professional challenge.
|NO ROOM FOR CLARKE AS RODGERS BUILDS AT ANFIELD
|New Liverpool manager reshuffles backroom leaving no room for assistant manager Steve Clarke
In the letter Rodgers looks back fondly on what he helped achieve at Swansea and tells the fans that they will continue to flourish without him.
He added: "My story at Swansea City began nearly two years ago and my mission above anything else was very clear: to inspire the city.
"My aim for a legacy was simply to have made Swansea City Football Club an attacking and creative team who were enjoyable to watch, and to make the people of Swansea proud of their great club, city and nation.
"In the years ahead, you, the most important people at the club, will decide if I achieved those dreams, but the journey for me has been one I will never forget.
"We created history together to become the first Welsh club in the Premier League, and defied all the odds at the same time, and there is no doubt the club will continue to flourish in the coming years."
Further praise is given to the fans whilst there is even a suggestion that a return to the club may be in the offering in the future.
He writes: "To the Jack Army. I’m proud to have been your manager and feel truly blessed for the support you have given me on and off the field.
"I leave knowing I couldn’t have given any more to help your great club succeed.
|TOUGH TIMES AHEAD FOR SWANS
|2/1||Swansea are 2/1 with Paddy Power to be relegated from next year's Premier League|
"Swansea will live in my heart for the rest of my life and one day I would love to return to this great club and city.
"Until that day comes, I wish everyone at Swansea lots of success in the next chapter of your amazing story."
Looking ahead to the future the new reds manager has received the backing of Liverpool keeper Jose Reina.
He told the Press Association: "His team has played well this year with an ethos that I share and Liverpool is also known to play the passing game but it is easier said than done.
"Rodgers is welcomed and I am at his disposal. We have won a trophy this year but it is clear that in recent seasons we should have done better."