On the existence of ratio limits of weighted $n$-generalized Fibonacci   sequences with arbitrary initial conditions
  We study ratio limits of the consecutive terms of weighted $n$-generalized Fibonacci sequences generated from arbitrary complex initial conditions by linear recurrences with arbitrary complex weights. We prove that if the characteristic polynomial of such a linear recurrence is asymptotically simple, then the ratio limit exists for any sequence generated from arbitrary nontrivial initial conditions and it is equal to the unique zero of the characteristic polynomial. 
Igor Szczyrba
math.NT