calculus and analysis - I want to be able to define (a(x) d/dx + b(x))^n

I want to be able to define an operator

$(a(x) d/dx + b(x))^n$

where $d/dx$ is the derivative operator and $a(x)$ and $b(x)$ are known functions and $n$ is a positive integer.

Related Query: How about defining

$ \prod_{i=1,...,n} (a_i(x) d/dx + b_i(x) ) $

where $a_i(x)$ and $b_i(x)$ are known functions and $n$ is a positive integer ?

Answer

This can be done with Nest:

abderiv[n_] = 
  Function[f, Nest[(a[x] D[#, x] + b[x]) &, f, n]];


abderiv[0][f[x]]

(* ==> f[x] *)

abderiv[1][f[x]]

(* ==> b[x] + a[x] Derivative[1][f][x] *)

abderiv[2][f[x]]


(*
==> b[x] + 
 a[x] (Derivative[1][b][x] + Derivative[1][a][x] Derivative[1][f][x] +
     a[x] (f^\[Prime]\[Prime])[x])
*)

Here the order n is provided as the argument to a function that is itself a function acting on an arbitrary expression f. The assumption in the question seems to be that the differentiation variable is always x, so I don't specify this variable as an additional argument.