Computing powers of the operator using symbolic computation

Question

Suppose $t\in\mathbb{R_+}$ - some parameter, $V: \mathbb{R}\to\mathbb{R}$ - some function. I have an operator $S:f\mapsto S[f]$ that maps a function $f$ to a function $S[f]$: $$ S[f](x) = f(x+\sqrt{t})+2f(x)+f(x-\sqrt{t})-\arctan(tV(x))f(x) $$ My goal is to find a formula for power $S^n$ of the operator $S$, and I wonder if Mathematica can help here. To find a general formula I first decided to compute some of the powers $S^2, S^3,...$ and look for a pattern. But doing so on paper is very tedious process. How can I input this formula to Mathematica and use its ability of symbolic computation to get formal representation (with like terms collected) for $S^2[f]=S[S[f]]$, $S^3[f]=S[S[S[f]]]$?

For example, formula for $S^2$ would look like this: $$ f(x+2\sqrt{t})+4f(x+\sqrt{t})+6f(x)+4f(x-\sqrt{t})+f(x-2\sqrt{t})-\arctan(tV(x+\sqrt{t}))f(x+\sqrt{t}) -4\arctan(tV(x))f(x)-\arctan(tV(x-\sqrt{t}))f(x-\sqrt{t}))-\arctan(tV(x))f(x+\sqrt{t})-\arctan(tV(x))f(x-\sqrt{t})+\arctan(tV(x))^2 $$

Simpler example: let $K[f][x] = f[x+t]+f[x]$. Then $$K^2[f][x] = K[f][x+t]+K[f][x] = f[x+2t]+2f[x+t]+f[x]$$ $$K^3[f][x] = K^2[f][x+t]+K^2[f][x] = f[x+3t]+3f[x+2t]+3f[x+t]+f[x]$$

So by inputing $K[f][x] = f[x+t]+f[x]$ and 3-rd power I would like to get the output $$f[x+3t]+3f[x+2t]+3f[x+t]+f[x]$$

Answer 1

I think it perhaps would help if you showed exactly what you expect by e.g. nesting your operator at least twice, so we can check the correctness of the results, but hopefully the following should get you started is going to work for you.

This is a simpler example to show how this approach might work:

ClearAll[op, f, a, v, t, x]
op[f_] = f[2 #] + v[t] &;
Through[NestList[op, a, 3][x]]

(* Out:
{a[x], a[2 x] + v[t], a[4 x] + 2 v[t], a[8 x] + 3 v[t]}
*)

Applying this to your problems will yield:

ClearAll[f, S, t, x]
S[f_] = (f[# + Sqrt[t]] + 2 f[#] + f[# - Sqrt[t]]) - ArcTan[t V[#]] f[#] &;
Through[NestList[S, f, 3][x]] // Simplify

(* Out:
{ f[x], 
  -(-2 + ArcTan[t V[x]]) f[x] + f[-Sqrt[t] + x] + f[Sqrt[t] + x], 
  (6 - 4 ArcTan[t V[x]] + ArcTan[t V[x]]^2) f[x] + f[-2 Sqrt[t] + x] + 4 f[-Sqrt[t] + x] 
  - ArcTan[t V[x]] f[-Sqrt[t] + x] - ArcTan[t V[-Sqrt[t] + x]] f[-Sqrt[t] + x] + 
  4 f[Sqrt[t] + x] - ArcTan[t V[x]] f[Sqrt[t] + x] - ArcTan[t V[Sqrt[t] + x]] f[Sqrt[t] + x] 
  + f[2 Sqrt[t] + x]}
*)

Answer 2

Update

This update is with code for the definition of the simpler function $K$.

ClearAll[K, PowerK]
K[f_, t_][x_] := f[x + t] + f[x];
PowerK[f_, t_, 1][x_] := K[f, t][x];
PowerK[f_, t_, 0][x_] := 1;
PowerK[f_, t_, n_][x_] := 
  PowerK[f, t, n - 1][x + t] + PowerK[f, t, n - 1][x];
Unprotect[Power];
Power[K, n_Integer][f_, t_][x_] := PowerK[f, t, n][x];

Here is how to use with Power:

(K^3)[f, t][x]
(* f[x] + 3 f[t + x] + 3 f[2 t + x] + f[3 t + x] *)

Here is an image with more examples:

First answer

Definition of $S$ and $f$:

S[f_, t_][x_] := f[x - Sqrt[t]] + 2 f[x] + f[x - Sqrt[t]];

f[x_] := x^3;

Note that $S$ in the question had several global parameters $t$ and $V$. I used only $t$.

Now we apply Nest:

Nest[S[f, 4], 1, 4]
(* -31318496301456 *)

Here is an image with more examples: