Abstract

This paper discusses the location of zeros of polynomials in a polynomial sequence \(\{P_{n}(z)\}_{n=1}^{\infty}\) generated by a three-term recurrence relation of the form \(P_{n}(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0\) with \(k>2\) and the standard initial conditions \(P_{0}(z)=1\), \(P_{-1}(z)=P_{-k+1}(z)=0,\) where \(A(z)\) and \(B(z)\) are arbitrary coprime real polynomials. We show that there always exist polynomials in \(\{P_{n}(z)\}_{n=1}^{\infty}\) with nonreal zeros.

中文翻译：

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满足三项递归关系的多项式序列中多项式的非实零

摘要

本文讨论了由形式为三项的递归关系生成的多项式序列\（\ {P_ {n}（z）\} _ {n = 1} ^ {\ infty} \）中多项式零的位置\（P_ {n}（z）+ B（z）P_ {n-1}（z）+ A（z）P_ {nk}（z）= 0 \）和\（k> 2 \）和标准初始条件\（P_ {0}（z）= 1 \），\（P _ {-1}（z）= P _ {-k + 1}（z）= 0，\）其中\（A（z）\ ）和\（B（z）\）是任意的互质实数多项式。我们证明\（\ {P_ {n}（z）\} _ {n = 1} ^ {\ infty} \）中始终存在带有非零的多项式。