Thanks for reply .. but I'd like to have this function:

Now .. How can I refer to the function s?

I want to print just this function..

Thanks!

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".

Mathematica is allergic to some big letters.

  eqs = {
  s == t*s + t*a + t*b + 1,
  a == t*c + t,
  b == t*d + t*f,
  c == t*s + t,
  d == t*s + t*a,
  f == t*c,
  e == t*d};
  FullSimplify@First@Solve[eqs, {a, b, c, d, e, f, s}, Reals] // 
    Column // TraditionalForm