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Correction to: On the spectral norms of r-circulant matrices with the bi-periodic Fibonacci and Lucas numbers

  • The original article was published in Journal of Inequalities and Applications 2017 2017:192

Correction

In the publication of this article [1], there are a few errors.

  1. (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. (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. (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. (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. (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. (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

  1. 1.

    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

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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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Correspondence to Yasin Yazlik.

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The online version of the original article can be found under https://doi.org/10.1186/s13660-017-1466-0.

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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

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