(q^(k-1)+1,q^k)型Fibonacci数列与Lucas数列的一个性质——兼谈二项式定理的一个推广

A Property of Fibonacci Sequence and Lucas Sequence of (q^(k-1)+1,q^k) Style——And Talk about an Extension of Binomial Theorem

多边形数、递推数列、二项式定理是初等数学的基本内容之一,素材丰富,独立成章.文[1]曾给出了二项式定理与等差数列的相关结论,但三者相互联系的研究成果甚少.本文讨论了它们的关联关系,将多边形数、一类递推数列巧妙地结合在推广的二项式定理之中.

One of the basic contents of elementary mathematics is polygonal number, recurrence sequence and binomial theorem, with rich material and is independent from each other. The results of binomial theorem and arithmetic sequence are given in the paper 1. But there are few research achievements. In this paper, their incidence relation is discussed, and polygonal number and a class of recurrence sequence are combined ingeniously in the extension of binomial theorem.