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Here is a way to apply a polynomial P in a differential operator D to a function f.

def polydop(P,D,f):
    ff = f
    g(t) = 0
    for c in P.coeffs():
        g = (g + c * ff).expand()
        ff = D(ff).expand()
    return g

To illustrate it on your example, define

• the differential operator:

  sage: def D(f):
  ....:    return t*f.derivative(t)
  

• the polynomial (in a suitable polynomial ring):

  sage: R = PolynomialRing(QQ,'x')
  sage: x = R.gen()
  sage: P = prod(x-n for n in xrange(1,6))
  

• and the function:

  sage: f(t,z) = z*t*exp(t/z)
  

and get the following result:

sage: g = polydop(P,D,f)
sage: g
(t, z) |--> t^6*e^(t/z)/z^4