关于Lucas序列的单调性

The monotonicity of Lucas sequences

设A和B是不等于 0的实数 ,Lucas序列 {un}和 {vn}满足递归关系 :u0 =0 ,u1=1,un+2 =Aun+1-Bun(n∈N) ;v0 =2 ,v1=A ,vn+2 =Avn+1-Bvn(n∈N) 本文确定了序列 {un}和 {vn}单调递增的充分必要条件 ,并用此结论得出了当m ,n为非负整数 ,A ,B为互素的非零整数且A2 ≥ 4B时 ,um(A ,B) ｜un(A ,B) ,vm(A ,B) ｜vn(A .B)

Let A and B be two nonzero real numbers,｛u n｝ and ｛v n｝ satisfy the recursive relation:u 0=0,u 1=1,u n+1 =Au n-Bu n+1 (n∈N);v 0=2,v 1=A,v n+2 =Av n+1 -Bv n(n∈N).In this paper,we determine the necessary and sufficient conditions that ｛u n｝ and ｛v n｝ are monotonic increasing.When A,B∈Z\｛0｝,(A,B)=1,A 2≥4B and m,n∈N,we also give the necessary conditions for u m(A,B)｜u n(A,B) and v m(A,B)｜v n(A,B).