切比雪夫序列初值精度有限时退化为周期序列

Chebyshev Sequences Degenerating into Periodic Sequences with Realizing Finite Precision Initial Value

深入研究并分析切比雪夫序列的短周期现象,并通过数学证明,对于任意整数阶切比雪夫序列,在初值角度取有限小数时必会退化为短周期序列.首先,通过证明得到切比雪夫序列退化为短周期序列的等价条件;然后,利用分类讨论的思想证明任意整数阶切比雪夫序列在初值角度取整数时,必会退化为短周期序列;最后,通过数学归纳法将这一结论扩展到任意有限精度初值上,得出切比雪夫序列在初值精度有限时退化为周期序列.

The short-period problems of Chebyshev sequences had been researched, and aconclusion that Chebyshev sequences of random integer order will degenerateinto short-period sequences while Chebyshev sequences＇ initial value is afinite decimal had been proved. The equivalent conditions about Chebyshevsequences degenerating into short-period sequences had been shown first, andthen an integer order Chebyshev sequences whose initial value is integermust degenerate into short-period sequence has been proved. Finally, theinitial value conditions has been extended to finite decimal by usingmathematical induction method.