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一、引言笔者曾在一个存在性问题的研究中,偶然地引出了如下一个由递推关系给出的多项式序列{f_n(x)}: f_o(x)=1,f_1(x)=r, f_n(x)=xf_(n-1)(x)-f_(n-2)(x),(n≥2)(1) 尽管其存在性问题早已解决,但由此多项式序列又意外地得到了几个有趣的组合恒等式以及一系列三角恒等式,同时还发现了一类三角函数式的求值方法。故书拙文,以求同行斧正, 二、f_n(x)的表达式与f_n(x)的根由于f_n(x)是x的多项式,因而自然地想求出它的表达式,容易用数学归纳法证明下面的定理1 对任意非负整数n,有其中[t]表示不超过实数t的最大整数。(证略) 当n≥5时,n次多项式的根无公式解,因
I. INTRODUCTION In a study of an existential problem, the author occasionally elicited the following polynomial sequence {f_n(x)} given by a recurrence relation: f_o(x)=1, f_1(x)=r, F_n(x)=xf_(n-1)(x)-f_(n-2)(x),(n≥2)(1) Although its existence problem has long been solved, the polynomial sequence is unexpectedly obtained. Several interesting combinatorial identities and a series of trigonometric identities were also found. At the same time, a trigonometric function evaluation method was also found. Therefore, the root of the expression f_n(x) and the root of f_n(x) are f_n(x) is a polynomial of x, and thus naturally want to find its expression, easy to use mathematical induction The proof of the following theorem 1 is for any non-negative integer n, where [t] represents the largest integer not exceeding the real number t. (Filing) When n≥5, the root of the polynomial of degree n has no formula solution.