Paktiv[n_, x_] = 
      Collect[(1/(n + 1) ((2  n + 1) x P[n, x] - 
                  n P[n - 1, x])) /. n -> n - 1, P[_], 
        Factor]

P[0, x] = 1
P[1, x] = x

P[2, x_] = Paktiv[2, x]

P[3, x_] = Paktiv[3, x]

P[4, x_] = Paktiv[4, x]

{Frage[2], Frage[3], Frage[4]} /. Derivative[0, 1][P][n_, x_] :> 
          D[P[n, x], x] // Simplify

Clear[Frage]

Frage[n_] = (x^2 - 1) D[P[n, x], x] - n x P[n, x] + 
        n P[n - 1, x]

Frage[2] /. 
          P[2, x] -> Paktiv[2, x] /. Derivative[0, 1][P][2, 
            x] -> D[Paktiv[2, x], x] // Simplify

e1 = 
      Simplify[Frage[n] /. 
            P[n, x] -> Paktiv[n, x] /. Derivative[0, 1][P][n, 
              x] -> D[Paktiv[n, x], x]]

e2 = e1 - richtig1 Frage[n - 1]

r2=Solve[Coefficient[e2,Derivative[0, 1][P][-1 + n, x]]==0,richtig1][[1]]

e3 = e2 /. r2 // Simplify

e4 = e3 - richtig2 Frage[n - 2]

r4=Solve[Coefficient[e4,Derivative[0, 1][P][-2 + n, x]]==0,richtig2][[1]]

e5 = e4 /. r4 // Simplify

e5 /. P[n - 1, x] -> Paktiv[n - 1, x] // Simplify