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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


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


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:

$$\begin{aligned}& \bigl({}_{k}^{s}D_{a+}^{\mu}f\bigr) (x)= \biggl[ \biggl(\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{n} \biggr] \bigl(k^{n}\ {}_{k}^{s}I_{a+}^{nk-\mu}f \bigr) (x), \end{aligned}$$
$$\begin{aligned}& \bigl({}_{k}^{s}D_{a-}^{\mu}f\bigr) (x)= \biggl[ \biggl(-\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{n} \biggr] \bigl(k^{n}\ {}_{k}^{s}I_{a-}^{nk-\mu}f \bigr) (x), \end{aligned}$$


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:

$$\begin{aligned} & \biggl(\frac{1}{x^{s}}\frac{d}{dx} \biggr)^{m} \bigl[\bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\beta }{k}-1}E_{k,\rho,\beta}^{\delta} \bigl(\omega\bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\rho }{k}} \bigr)\bigr] \\ &\quad =\frac{(s+1)^{m}(x^{s+1}-a^{s+1})^{\frac{\beta}{k}-m-1}}{k^{m}}E_{k,\rho ,\beta-mk}^{\delta}\bigl(\omega \bigl(x^{s+1}-a^{s+1}\bigr)^{\frac{\rho}{k}}\bigr), \end{aligned}$$

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.


  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).

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