Abstract

In this talk I′m going to consider the $k$-order rational difference equation

 un+k=(A1un+A2un+1+...+ Akun+k-1+A0)/ (B1un+B2un+1+...+ Bkun+k-1+B0) 

with positive initial conditions and
$\sum$_i=0^kA_i>0, ∑_i=0^kB_i>0, A_i≥0, B_i≥0.

The problem which we study is the characterization of the difference equations of the above form such that there exists a natural $p$ for which $u$_n+p=u_n for all $n\ge 1$.

We give the determination of the $k$-order $p$-periodic rational recurrences for small values of $k$. At the same time I′m going to explain a general result of Csörnyei and Laczkovich about de same problem.