OrePolynomial[{{t^3 - t^5 - t^7 + t^9, {4}}, {4*(-2*t^4 - t^6 + 3*t^8), {3}}, 
  {3*(-t - 3*t^3 + t^5 + 11*t^7), {2}}, {3*(1 + 3*t^2 - t^4 + 5*t^6), {1}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
