Clear[Delta]
Delta[f_, k_] := (f /. k -> k + 1) - f

Delta[(k^3), k]

Delta[(k^3), k] // Expand

Clear[fallendeFaktorielle]
fallendeFaktorielle[k_, j_] := Pochhammer[k - j + 1, j]

Delta[fallendeFaktorielle[k, 3], k]

Delta[fallendeFaktorielle[k, 3], k] // Factor

fallendeFaktorielle[k, 3] + 3  fallendeFaktorielle[k, 2] + 
        fallendeFaktorielle[k, 1] // Expand

Clear[Summation]
Summation[c_, k_] := 
        c k /; FreeQ[c, k];
Summation[c_ f_, k_] := 
      c Summation[f, k] /; FreeQ[c, k]
Summation[f_ + g_, k_] := 
      Summation[f, k] + Summation[g, k]
Summation[FallendeFaktorielle[k_, n_], k_] := 
      FallendeFaktorielle[k, n + 1]/(n + 1) /; 
        n ≥ 0
\(Summation[k_\^n_. , k_] := 
      Module[{j}, 
        Expand[Summation[FallendeFaktorielle[k, n], k] + 
            Summation[
              Expand[k\^n - ∏\+\(j = 0\)\%\(n - 1\)\((k - j)\)], 
              k]]]\)

Summation[(k^3), k]

s = 
      Summation[(k^3), k] /. FallendeFaktorielle -> fallendeFaktorielle // 
        Factor

(s /. k -> n + 1) - (s /. k -> 1)