algebra of differential operators

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I've attached some code below that defines your operator in a way that lets it operate on polynomials (it is the operator "op" after the code is run). However, I'm pretty new to Sage, so I'm not sure how to make it act on exponential functions. (Also, since I'm new to Sage, this might be a bad way to make this operator, but it does seem to work, at least.)

RR = PolynomialRing(QQ,'t')
t = RR.gens()[0]

id = lambda x : x
def comp(a,b):
    return lambda x: a(b(x))

def Dtn(n):
    return lambda f: t*f.derivative(t) - n*f

op = id
for n in range(5):
    optmp = op
    op = comp(optmp,Dtn(n+1))

Indeed, python functions like Df do not like to be added to integers like n-1, or multiplied together.

Is there a reason not to do the computation in a single step ?

sage: var('t z')
sage: f = z*t*exp(t/z)
sage: prod([ t*f.derivative(t) - n-1 for n in range(5) ])
((t*e^(t/z) + z*e^(t/z))*t - 1)*((t*e^(t/z) + z*e^(t/z))*t - 2)*((t*e^(t/z) + z*e^(t/z))*t - 3)*((t*e^(t/z) + z*e^(t/z))*t - 4)*((t*e^(t/z) + z*e^(t/z))*t - 5)

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