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

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.