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

The Original Article was published on 21 April 2017

Abstract

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

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.

References

  1. 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:10.1186/s13662-017-1176-4

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Correspondence to KS Nisar.

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Nisar, K., Rahman, G., Baleanu, D. et al. Erratum to: The \((k,s)\)-fractional calculus of k-Mittag-Leffler function. Adv Differ Equ 2017, 194 (2017). https://doi.org/10.1186/s13662-017-1239-6

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