Problem with using RSolve to solve recurrence equations

Question

Whenever I run the my code, the output of Mathematica is only a copy of the input again.

RSolve[{x[1 + k] == (-(Cos[(k π)/2]/k!) + z[k])/(1 + k), 
y[1 + k] == (1/k! + z[k])/(1 + k), 
z[1 + k] == (x[k] - y[k])/(1 + k), 
x[0] == x0, y[0] == y0, z[0] == z0}, {x[k], y[k], z[k]}, k]

I cannot find what I am doing wrong with my simple code.

Answer 1

You can use the first two equations to get an algebraic equation for z[k]:

(* x[k+1] - y[k+1] then compare to third equation *)
zksol = Simplify[(-(Cos[(k π)/2]/k!) + z[k])/(1 + k) - (1/k! + 
   z[k])/(1 + k) /. k -> (k - 2), Assumptions -> {k ∈ Integers}] 
(* -((2 Sin[(k π)/4]^2)/((-1 + k) (-2 + k)!)) *)

Then you can solve a smaller system

{xksol[k_], yksol[k_]} = {x[k], y[k]} /. 
First@RSolve[{x[1 + k] == (-(Cos[(k π)/2]/k!) + zksol)/(1 + k), 
             y[1 + k] == (1/k! + zksol)/(1 + k)}, {x[k], y[k]}, k]
    (* {(2 Cos[1/2 (-1 + k) π] (-3 + k)! - 
  k Cos[1/2 (-1 + k) π] (-3 + k)! - (-1 + k)! + 
  Cos[1/2 (-1 + k) π] (-1 + k)!)/((-2 k + 
    k^2) (-3 + k)! (-1 + k)!), (-2 (-3 + k)! + 
  k (-3 + k)! - (-1 + k)! + 
  Cos[1/2 (-1 + k) π] (-1 + k)!)/((-2 k + 
    k^2) (-3 + k)! (-1 + k)!)} *)

Check:

Simplify[{x[1 + k] == (-(Cos[(k π)/2]/k!) + z[k])/(1 + k), 
         y[1 + k] == (1/k! + z[k])/(1 + k)} /. 
         {x[p_] -> xksol[p], y[p_] -> yksol[p], z[k] -> zksol}]
(* {True, True} *)

Notice there is no freedom left for initial conditions.