单位圆上的交错：通过系数倒数
Interlacing on the Unit Circle via Coefficientwise Reciprocals
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中文总结 AI 辅助
本文证明具有负实数零点的回文多项式之系数倒数在单位圆上严格交错，基于有限Blaschke乘积，并应用于二项式、欧拉、Narayana等多项式族。
中文摘要 AI 辅助
设 $f(z)=\sum_{k=0}^{n}a_kz^k$ 为具有正系数的多项式，并定义其系数倒数 $f^{\\#}(z)=\sum_{k=0}^{n}\frac{z^k}{a_k}.$ 已知若 $f$ 是回文的且仅有负实数零点，则 $f^{\\#}$ 的所有零点位于单位圆上。我们证明：若 $p$ 和 $q$ 分别是次数为 $n$ 和 $n+1$ 的回文多项式，且仅有负实数零点，则它们的系数倒数仅有单零点，并且 $p^{\\#}$ 在单位圆上严格交错于 $q^{\\#}$。我们的证明基于有限 Blaschke 乘积及其边界相位的比较。作为直接推论，我们得到了倒数二项式、倒数欧拉和倒数 Narayana 多项式的严格交错性。通过将 $\gamma$-系数的符号变化与系数倒数相结合，我们进一步从 Rogers–Szegő、Poupard 和 Kreweras 相关多项式构造了严格交错族。
英文摘要
Let $f(z)=\sum_{k=0}^{n}a_kz^k$ be a polynomial with positive coefficients, and define its coefficientwise reciprocal by $f^{\#}(z)=\sum_{k=0}^{n}\frac{z^k}{a_k}.$ It is known that if $f$ is palindromic and has only negative real zeros, then all zeros of $f^{\#}$ lie on the unit circle. We prove that if $p$ and $q$ are palindromic polynomials of degrees $n$ and $n+1$, respectively, with only negative real zeros, then their coefficientwise reciprocals have only simple zeros, and $p^{\#}$ strictly interlaces $q^{\#}$ on the unit circle. Our proof is based on finite Blaschke products and the comparison of their boundary phases. As an immediate consequence, we obtain strict interlacing for the reciprocal binomial, reciprocal Eulerian, and reciprocal Narayana polynomials. By combining a sign change in the $γ$-coefficients with coefficientwise reciprocation, we further construct strictly interlacing families from Rogers--Szegő, Poupard, and Kreweras-related polynomials.
发表机构
- Dalian University of Technology（大连理工大学）
- Dongbei University of Finance and Economics（东北财经大学）
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