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.
References
Köme, C., Yazlik, Y.: On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers. J. Inequal. Appl. 2017(1), 192 (2017). https://doi.org/10.1186/s13660-017-1466-0
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.
Author information
Authors and Affiliations
Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that there are no competing interests with any individual or institution.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The online version of the original article can be found under https://doi.org/10.1186/s13660-017-1466-0.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Köme, C., Yazlik, Y. Correction to: On the spectral norms of r-circulant matrices with the bi-periodic Fibonacci and Lucas numbers. J Inequal Appl 2018, 50 (2018). https://doi.org/10.1186/s13660-018-1642-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1642-x