一类Lyness方程解的性质

Properties of Solution in A Class Lyness Equation

考虑差分方程xn+1=a+b0xn+b1xn-1+…+bk-1xn-(k-1)xn-k其中a,bi是非负实数,a+∑k-1i=0bi>0,k∈{1,2,…}.证明了当k+1为素数时,方程的任半环不超过(2k+2)项;当k+1为合数且只有一个bi≠0时,方程的任半环不超过2k+1+km+0 1项,其中m0=min{m m为k+1的大于1的因数}.结果部分回答了C.Darwen and W.T.Patula提出的公开问题.

In this paper, we study further on the cycle length for the difference equation xn＋1=xn-k/（a＋∑i=0 ^k-1 bixn-i） where a and bi are nonnegative numbers and （a＋∑i=0 ^k-1 bi）〉 0. It is showed that if k＋1 is a prime number, then every semicycle has no more than （2k ＋ 2） terms; if k ＋ 1 is a composite number and only one bi≠0, then every semicycle has no more than （ 2k ＋ 1 ＋ m0/k＋1） . where m0 = min{m /m〉 1 ,is a factor of k ＋ 1}. This result partially answer the open question posed by C. Darwen and W. T. Patula.