How to calculate an iterated derivative in Mathematica?

Question

I try to calculate an inverse Mellin transform for $s^n \Gamma(s)$: $$x\frac{\mathrm{d}}{\mathrm{d}x}\left(x\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}\right)\right)$$ for $n=2$ for $n$ as $$\left(x\text{}\frac{\partial }{\partial x}\right)^ne^{-x}$$ I try NestList and Do but it does not work; it is simple but I don't know how to do it.

Answer 1

You are basically differentiating with respect to Log[x], which is possible using the ResourceFunction "ChainD":

ResourceFunction["ChainD"][Exp[-x],{Log[x],2}]

-E^-x x + E^-x x^2

Answer 2

One way is:

f[n_] := If[n == 0, E^-x, (-1)^n*Nest[x D[#, x] &, x D[Exp[-x], x], n - 1] // Simplify]
(for n >= 0 and n is integer )
f[2]
(E^-x (-1 + x) x)
Grid[{Table[f[n], {n, 0, 5}], Table[InverseMellinTransform[s^n Gamma[s], s, x], {n, 0, 5}]}, Frame -> All]

Answer 3

$Version

(* "12.2.0 for Mac OS X x86 (64-bit) (December 12, 2020)" *)

Clear["Global`*"]

There are multiple ways of implementing the differential operator

dOp1[f_, n_Integer?NonNegative, sym : _Symbol : x] :=
 Nest[Simplify[sym*D[#, sym]] &, f, n]
dOp2[f_, n_Integer?NonNegative, sym : _Symbol : x] :=
 Sum[StirlingS2[n, k]sym^kD[f, {sym, k}], {k, 0, n}]
dOp3[f_, n_Integer?NonNegative, sym : _Symbol : x] :=
 D[f /. sym :> E^sym, {sym, n}] /. sym :> Log[sym]

Comparing the implementations

And@@Table[dOp1[f[x], n] == dOp2[f[x], n] == dOp3[f[x], n], {n, 0, 15}] // 
  Simplify
(* True *)

For the special case of f[x] == E^-x

And@@With[{f = E^-x}, 
 Table[dOp1[f, n] == dOp2[f, n] == dOp3[f, n] == 
    (-1)^n*InverseMellinTransform[s^n Gamma[s], s, x], {n, 0, 15}] // 
  Simplify]
(* True *)