Open Access

Erratum to: The \((k,s)\)-fractional calculus of k-Mittag-Leffler function

Advances in Difference Equations20172017:194

https://doi.org/10.1186/s13662-017-1239-6

Received: 12 June 2017

Accepted: 12 June 2017

Published: 7 July 2017

The original article was published in Advances in Difference Equations 2017 2017:118

Abstract

In this note we present some corrections to our previous paper (Nisar et al. in Adv. Differ. Equ. 2017:118, 2017).

1 Erratum

In the paper [1], the following errors are present on pages 4, 5, 6 and 7.

In Definition 3, in equations (20) and (21), one left bracket is misplaced inside the expression \([ (\frac{1}{x^{s}}\frac{d}{dx} )^{n} ]\) and \([ (-\frac{1}{x^{s}}\frac{d}{dx} )^{n} ]\), respectively. The correct forms of the expressions are as follows:
$$\begin{aligned}& \bigl({}_{k}^{s}D_{a+}^{\mu}f\bigr) (x)= \biggl[ \biggl(\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{n} \biggr] \bigl(k^{n}\ {}_{k}^{s}I_{a+}^{nk-\mu}f \bigr) (x), \end{aligned}$$
(1)
$$\begin{aligned}& \bigl({}_{k}^{s}D_{a-}^{\mu}f\bigr) (x)= \biggl[ \biggl(-\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{n} \biggr] \bigl(k^{n}\ {}_{k}^{s}I_{a-}^{nk-\mu}f \bigr) (x), \end{aligned}$$
(2)
respectively.
On page 5, in the proof of Lemma 1, line 6, the numerator confuses \((1-\mu)\) and \((k-\mu)\), the correct expression is
$$\begin{aligned} &\frac{1}{x^{s}}\frac{d}{dx} \bigl({}_{k}^{s}I_{a+}^{(1-\nu)(k-\mu )} \bigl[\bigl(t^{s+1}-a^{s+1}\bigr)^{\frac{\lambda}{k}-1}\bigr] \bigr) (x) \\ &\quad =\frac{[(1-\nu)(k-\mu)+\lambda-k]\Gamma_{k}(\lambda)}{ k(s+1)^{\frac{(1-\nu)(k-\mu)}{k}-1}\Gamma_{k}((1-\nu)(k-\mu)+\lambda )}\bigl(x^{s+1}-a^{s+1} \bigr)^{\frac{(1-\nu)(k-\mu)+\lambda}{k}-2}. \end{aligned}$$
On page 6, Theorem 1, equation number (24) is misplaced and now equation (25) is (24) (accordingly, all equation numbers will change). In the statement of Theorem 1 at the beginning \(\frac{1}{x^{\frac {s}{m}}}\) should instead read \(\frac{1}{x^{s}}\). Also the power \(\frac {c}{k}\) should instead read \(\frac{\beta}{k}\). The correct expression is as follows:

Theorem 1

For \(k>0\), the following result always holds true:
$$\begin{aligned} & \biggl(\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{m} \bigl[\bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\beta }{k}-1}E_{k,\rho,\beta}^{\delta} \bigl(\omega\bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\rho }{k}} \bigr)\bigr] \\ &\quad =\frac{(s+1)^{m}(x^{s+1}-a^{s+1})^{\frac{\beta}{k}-m-1}}{k^{m}}E_{k,\rho ,\beta-mk}^{\delta}\bigl(\omega \bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\rho}{k}}\bigr), \end{aligned}$$
(3)
where \(s\in\mathbb{R}\backslash\{-1\}\), \(\mu, \rho, \beta, \delta\in \mathbb{C}\), \(\Re(\mu)>0\) and \(\Re(\beta)>0\), \(\Re(\rho)>0\), \(\Re(\delta)>0\).

Also, in the proof of Theorem 1, the error: \(\frac{1}{x^{\frac{s}{m}}}\) should instead read: \(\frac{1}{x^{s}}\).

On page 7 in the proof of equation (27) (just after the sentences ‘This completes the proof of (26). Now, we have’ in the second line of the expression) the error: \((\frac{1}{x^{\frac{s}{n}}}\frac{d}{dx} )^{n}\) should instead read: \((\frac{1}{x^{s}}\frac{d}{dx} )^{n}\). Also (just after the sentences ‘and using (26) this takes the following form’ in the second line of the expression) the error: \((\frac{1}{x^{\frac {s}{n}}}\frac{d}{dx} )^{n}\) should instead read: \((\frac{1}{x^{s}}\frac{d}{dx} )^{n}\). This has now been included in this erratum.

Notes

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, College of Arts and Science - Wadi Aldawaser, Prince Sattam bin Abdulaziz University
(2)
Department of Mathematics, International Islamic University
(3)
Department of Mathematics, Cankaya University
(4)
Department of Mathematics, University of Sargodha

References

  1. Nisar, KS, Rahman, G, Baleanu, D, Mubeen, S, Arshad, M: The \((k,s)\)-fractional calculus of k-Mittag-Leffler function. Adv. Differ. Equ. 2017, 118 (2017). doi:https://doi.org/10.1186/s13662-017-1176-4 MathSciNetView ArticleGoogle Scholar

Copyright

© The Author(s) 2017