functions - Polynomial expansion of operator

I am new to Mathematica, I am trying to generate the polynomial function of a operator. So for example, the operator $L $ is $\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y} $, and I want to generate the polynomial $ \sum_{n=0}^{n=k} (L/2)^n$ and apply that polynomial operator to a function. I was trying to use Nest, any help??

Answer

Define

L = (1/2) (D[#, x] + D[#, y]) &

We see that L works as desired. For instance:

Simplify[Nest[L, f[x, y], 3]]
(* (Derivative[0, 3][f][x, y] + 3*Derivative[1, 2][f][x, y] + 3*Derivative[2, 1][f][x, y] +
   Derivative[3, 0][f][x, y])/8 *)

And the Sum can be constructed in a similar manner. For instance:

Simplify[Sum[Nest[L, f[x, y], n], {n, 0, 3}]]

(* (8*f[x, y] + 4*Derivative[0, 1][f][x, y] + 2*Derivative[0, 2][f][x, y] +
    Derivative[0, 3][f][x, y] + 4*Derivative[1, 0][f][x, y] + 
    4*Derivative[1, 1][f][x, y] + 3*Derivative[1, 2][f][x, y] + 2*Derivative[2, 0][f][x, y] +
    3*Derivative[2, 1][f][x, y] + Derivative[3, 0][f][x, y])/8 *)

Update

As kindly pointed out by @b.gatessucks in the Comment below, computation of the final result can be simplified with NestList. (Thanks!)

Simplify[Total[NestList[L, f[x, y], 3]]]

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