This is a very unclear question, both in terms of Mathematica and in the underlying mathematics. Here are a few suggestions.
(1) Pare it down to a smaller example.
(2) Explain clearly what is desired both for input and output.
(3) As part of item (2) make sure to explain carefully if this is intended to be a system of differential equations, difference equations, polynomial equations, or womething else entirely.
(4) As part of item (3), make sure to explain what you have in mind when mentioning "the s generating function".
