连续多项式、切比雪夫多项式和莱利多项式
Continuant, Chebyshev polynomials, and Riley polynomials
摘要：
在前面的文章中，我们证明了每个2-桥结$K$的Riley多项式$\mathcal{R}_K(λ)$分裂为$\mathcal{R}_K(-u^2)=\pm g(u)g(-u)$，对于某些整系数多项式$g(u)\in \mathbb Z[u]$。本文研究了Riley多项式的这种分裂性质。证明了Riley多项式可表示为“$ε$ -切比雪夫多项式”，它是含有2-桥纽结$K=S(α,β)$的$ε_i$ -序列$(ε_i=(-1)^{[i\fracβα]})$信息的切比雪夫多项式的推广，并给出了分裂多项式$g(u)$也可表示为$ε$ -切比雪夫多项式的显式.作为应用，我们得到了Riley多项式不可约的一个充分条件，并证明了对称化Riley多项式的单峰性。
英文摘要：
In the previous paper, we showed that the Riley polynomial $\mathcal{R}_K(λ)$ of each 2-bridge knot $K$ is split into $\mathcal{R}_K(-u^2)=\pm g(u)g(-u)$, for some integral coefficient polynomial $g(u)\in \mathbb Z[u]$. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by `$ε$-Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of $ε_i$-sequence $(ε_i=(-1)^{[i\fracβα]})$ of the 2-bridge knot $K=S(α,β)$, and then we give an explicit formula for the splitting polynomial $g(u)$ also as $ε$-Chebyshev polynomials. As applications, we find a sufficient condition for the irreducibility of the Riley polynomials and show the unimodal property of the symmetrized Riley polynomial.
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