I am not sure that I understand exactly what you are intended to do but you can pull the polynomial expression apart. First you temporarily replace the u by a variable v with positive integer exponents:
t3 = t2 /. Power[ u_, Rational[p_, m_]] -> Power[v, -(p/m + 1/2)]
Now you use CoefficientList to extract the coefficients:
t4 = CoefficientList[t3, {v}];
Finally you bring back u into a separate list. The result looks like
t5 = {v^Range[0, Length[t4] - 1] /. {Power[v, n_] ->
Power[u, -n - 1/2], v -> u^(-3/2)}, t4}
Here is a check that this decomposition corresponds in fact to T2
Total[Map[Apply[Times, #] &, t5\[Transpose]]] == t2
Hope that helps
