In Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. we have considered the problem about asymptotic formulas for all eigenvalues of $T_n(a)$, as n goes to infinity, assuming that a is a specific model symbol with a unique zero of order 4. In this paper, we continue our investigation and we explore the case where a is a more general real-valued rational symbol with a unique zero of order 4. It should be noted that we apply a different method than the one used in Barrera M, Grudsky SM. Asymptotics of eigenvalues for pentadiagonal symmetric Toeplitz matrices. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017; p. 51–77. This method coming from works Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols. J Math Anal Appl. 2015;422(2):1308–1334 and Bogoya JM, Böttcher A, Grudsky SM, et al. Eigenvalues of Hermitian Toeplitz matrices generated by simple-loop symbols with relaxed smoothness. In: Large truncated Toeplitz matrices, Toeplitz operators, and related topics. Operator theory: advances and applications Vol. 259, Birkhäuser, Cham.; 2017. p. 179–212, where it is considered the class of all symbols having zeros of second order and one can reduce the problem to asymptotic analysis of a nonlinear equation. As well, we construct uniform asymptotic expansions for all eigenvalues, which allow us to precise the classical results of Widom and Parter for first and very last eigenvalues.

在 Barrera M，Grudsky SM。五对角对称 Toeplitz 矩阵的特征值的渐近。在：大型截断 Toeplitz 矩阵、toeplitz 算子和相关主题。算子理论：进展与应用卷。259，伯克豪瑟，湛。2017；页。51-77。我们已经考虑了关于所有特征值的渐近公式的问题${吨}_n(\mathrm{一种})$，当 n 趋于无穷大时，假设a是具有唯一 4 阶零的特定模型符号。在本文中，我们继续研究并探讨a是一个更一般的实值有理符号，具有唯一的 4 阶零。应该注意的是，我们应用的方法与 Barrera M, Grudsky SM 中使用的方法不同。五对角对称 Toeplitz 矩阵的特征值的渐近。在：大型截断 Toeplitz 矩阵、Toeplitz 算子和相关主题。算子理论：进展与应用卷。259，伯克豪瑟，湛。2017；页。51-77。这种方法来自 Bogoya JM、Böttcher A、Grudsky SM 等人的作品。具有平滑单环符号的 Hermitian Toeplitz 矩阵的特征值。J数学肛门应用程序。2015;422(2):1308–1334 和 Bogoya JM、Böttcher A、Grudsky SM 等。Hermitian Toeplitz 矩阵的特征值由具有松弛平滑度的简单循环符号生成。在：大型截断 Toeplitz 矩阵、Toeplitz 算子和相关主题。算子理论：进展与应用卷。259，伯克豪瑟，湛。2017 年。179-212，其中它被认为是具有二阶零的所有符号的类别，并且可以将问题简化为非线性方程的渐近分析。同样，我们为所有特征值构造了一致的渐近展开，这使我们能够精确 Widom 和 Parter 的第一个和最后一个特征值的经典结果。