1 Correction
In the publication of this article [1], there are a few errors.
(1)
Page 4, line 4:
The statement \(\frac{1}{(ab)^{m+1}} [ \alpha^{2m+1} + \beta^{2m+1} - (-1)^{m} ] -2\) should instead read: \(\frac{1}{(ab)^{m+1}} [ \alpha^{2m+1} + \beta^{2m+1} ] + (-1)^{m} -2\).
(2)
Page 4, line 6:
The statement \(( \frac{1}{a} ) l_{m} l_{m+1} = \frac{1}{(ab)^{m+1}} [ \alpha^{2m+1} + \beta^{2m+1} - (-1)^{m} ] \) should instead read: \(( \frac{1}{a} ) l _{m} l_{m+1} = \frac{1}{(ab)^{m+1}} [ \alpha^{2m+1} + \beta^{2m+1} ] + (-1)^{m}\).
(3)
Page 8, Equation (16):
The matrix F should instead read:
$$ F = \begin{bmatrix} 1 & 1 & 1 & \ldots & 1 \\ r ( \frac{b}{a} ) ^{\frac{\xi {(n-1)}}{2}} l_{n-1} & 1 & 1 & \ldots & 1 \\ r ( \frac{b}{a} ) ^{\frac{\xi {(n-2)}}{2}} l_{n-2} & r ( \frac{b}{a} ) ^{\frac{\xi {(n-1)}}{2}} l_{n-1} & 1 & \ldots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ r ( \frac{b}{a} ) ^{\frac{\xi {(1)}}{2}} l_{1} & r ( \frac{b}{a} ) ^{\frac{\xi {(2)}}{2}} l_{2} & r ( \frac{b}{a} ) ^{\frac{\xi {(3)}}{2}} l_{3} & \ldots & 1 \end{bmatrix} . $$
(4)
Page 8, line 15: The equation \(r_{1}(F)\) should instead read:
$$ r_{1}(F) = \max_{1\leq i\leq n} \sqrt{\sum _{j=1}^{n}\vert f_{ij} \vert ^{2} } = \sqrt{1 + \vert r \vert ^{2}\sum _{k=1}^{n-1} \biggl( \frac{b}{a} \biggr) ^{\xi (k)} l_{k}^{2} } = \sqrt{1 + \vert r \vert ^{2} \biggl( \frac{l_{n} l_{n-1}}{a}-2 \biggr) }. $$
(5)
Page 9, lines 2 and 4, page 10, line 11 and Theorem 2.3 on page 7:
The statement \(\vert r \vert ( \frac{l_{n} l_{n-1}}{a} + 2 ) \) should instead read:
$$ \sqrt{\frac{l_{n} l_{n-1}}{a}+2} \sqrt{1 + \vert r \vert ^{2} \biggl( \frac{l _{n} l_{n-1}}{a}-2 \biggr) }. $$
(6)
Page 10, line 20:
The statement \(\vert r \vert ^{2} \frac{q_{n}q_{n-1}}{a} ( \frac{l_{n} l _{n-1}}{a} + 2 ) l\) should instead read:
$$ \vert r \vert \frac{q_{n}q_{n-1}}{a} \sqrt{\frac{l_{n} l_{n-1}}{a}+2} \sqrt{1 + \vert r \vert ^{2} \biggl( \frac{l_{n} l_{n-1}}{a}-2 \biggr) }. $$
This has now been included in this erratum.
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Acknowledgements
The authors are grateful to the anonymous referees who have contributed to improve the quality of the paper. The authors declare that they have not received any financial support to do this research.
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