arXiv 1503.054042015-03-10math.CAmath.CO

多项式序列II的根几何：（1，0）型

Root geometry of polynomial sequences II: Type (1,0)

J. L. Gross, T. Mansour, T. W. Tucker, D. G. L. Wang

备注：: 37 pages

摘要：

我们考虑由递归$W_n(x)=(ax+b)W_{n-1}(x)+dW_{n-2}(x)$定义的多项式序列$W_n(x)$，初始值为$W_0(x)=1$和$W_1(x)=t(x-r)$，其中$a,b,d,t,r$是实数，$a,t>0$和$d<0$。我们证明了每个多项式$W_n(x)$是异实根的，并且多项式$W_n(x)$的根与多项式$W_{n-1}(x)$的根交错。我们发现，作为$n\to\infty$，多项式$W_n(x)$的最小根序列递减地收敛到一个实数，最大根序列递增地收敛到一个实数。利用Dirichlet逼近定理，证明了对任意正整数$i\ge2$，多项式$W_n(x)$的$i$次最小根序列收敛于某个数.类似地，对于每个正整数$i\ge2$，多项式$W_n(x)$的第$i$个最大根的序列收敛到一个数。事实证明，这两个收敛点独立于数字$t$和$r$以及$i$。我们推导出这四个极限点的显式表达式，我们完全确定当这些极限点重合。

英文摘要：

We consider the sequence of polynomials $W_n(x)$ defined by the recursion $W_n(x)=(ax+b)W_{n-1}(x)+dW_{n-2}(x)$, with initial values $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a,b,d,t,r$ are real numbers, $a,t>0$, and $d<0$. We show that every polynomial $W_n(x)$ is distinct-real-rooted, and that the roots of the polynomial $W_n(x)$ interlace the roots of the polynomial $W_{n-1}(x)$. We find that, as $n\to\infty$, the sequence of smallest roots of the polynomials $W_n(x)$ converges decreasingly to a real number, and that the sequence of largest roots converges increasingly to a real number. Moreover, by using the Dirichlet approximation theorem, we prove that there is a number to which, for every positive integer $i\ge2$, the sequence of $i$th smallest roots of the polynomials $W_n(x)$ converges. Similarly, there is a number to which, for every positive integer $i\ge2$, the sequence of $i$th largest roots of the polynomials $W_n(x)$ converges. It turns out that these two convergence points are independent of the numbers $t$ and $r$, as well as $i$. We derive explicit expressions for these four limit points, and we determine completely when some of these limit points coincide.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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