# Note on generalized Mittag-Leffler function

## Abstract

The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemannâ€“Liouville fractional integration operator has been obtained.

## Background

The well-known Mittag-Leffler function $$E_{\alpha } (z)$$ named after its originator, the Swedish mathematician Gosta Mittag-Leffler (1846â€“1927), is defined by (Mittag-Leffler 1903)

$$E_{\alpha } (z) = \sum\limits_{n = 0}^{\infty } {\frac{{z^{n} }}{\Gamma (\alpha n + 1)}} ; \, \quad z{\text{ is a complex variable and}}\,\text{Re} \left( \alpha \right) \ge 0.$$
(1)

The Mittagâ€“Leffler function naturally occurs as the solution of fractional order differential equations. The various generalization of Mittagâ€“Leffler function have been defined and studied by different authors.

Shukla and Prajapati (2007) introduced its generalization $$E_{\alpha ,\beta }^{\gamma ,q} (z)$$, this is defined as

$$E_{\alpha ,\beta }^{\gamma ,q} (z) = \sum\limits_{n = 0}^{\infty } {\frac{{(\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}} \frac{{z^{n} }}{n!};$$
(2)

for $$\alpha ,\beta ,\gamma \in C$$; $$\text{Re} \left( \alpha \right) > 0,\,\text{Re} \left( \beta \right) > 0,\text{Re} \left( \gamma \right) > 0,\text{Re} \left( \delta \right) > 0$$, $$q \in (0,1) \cup N$$, and $$(\gamma )_{qn} = \frac{\Gamma (\gamma + qn)}{\Gamma (\gamma )}$$ denotes the generalized Pochhammer symbol.

Further, the generalization of (2) is also given by Khan and Ahmed (2013), as follows:

$$E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (z) = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} z^{n} }}{{\Gamma (\alpha n + \beta )(\upsilon )_{\sigma n} (\delta )_{pn} }}} ,$$
(3)

where $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon \in C$$; $$p,q,\rho ,\sigma > 0$$;$$q \le Re(\alpha ) + p$$; $$\rho \le \text{Re} \left( \sigma \right) + p$$; $$q \le \text{Re} \left( \sigma \right) + p$$; $$\rho ,q \in (0,1) \cup N$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right), \, \text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right)} \right) > 0.$$

Here, the convergence conditions of (3) have been modified, which was given by Khan and Ahmed (2013).

The following well-known notations and definitions have been used:

Let $$L(a,b)$$ (Kilbas et al. 2004) be a set of all Lebesgue measurable real or complex valued functions $$f(x)$$ on $$[a,b]$$ i.e.

$$L(a,b) = \left\{ {f:\left\| f \right\|_{1} \equiv \int\limits_{a}^{b} {\left| {f(t)} \right|dt < \infty } } \right\}$$
(4)

Let $$f(x) \in L(a,b)$$, $$\mu \in C$$ $$(\text{Re} (\mu ) > 0)$$ then the Riemannâ€“Liouville left-sided fractional integrals of order $$\mu$$ (MillerÂ and Ross 1993) is defined as

$${}_{a}I_{x}^{\mu } f(x) = I_{a + }^{\mu } f(x) = \frac{1}{\Gamma (\mu )}\int\limits_{a}^{x} {\frac{f(t)}{{(x - t)^{1 - \mu } }}} dt\,\,\,\,(x > a)$$
(5)

and the Râ€“L right-sided fractional integral of order $$\mu$$ is defined as

$$_{b} I_{x}^{\mu } f\left( x \right) = I_{b - }^{\mu } f\left( x \right) = \frac{1}{\Gamma (\mu )\,}\int\limits_{x}^{b} {\frac{f(t)}{{(t - x)^{1 - \mu } }}} dt\,\,\,(x < b)$$
(6)

Miller and Ross (1993) defined the following:

If $$\mu ,\alpha ,\beta \in C$$, $$\text{Re} (\mu ) > 0$$; $$n = [\text{Re} (\mu )] + 1$$; $${\text{Re}} (\beta ) > 0$$ then

$$I_{a + }^{\mu } [(t - a)^{\beta - 1} ](x) = \frac{\Gamma (\beta )}{\Gamma (\mu + \beta )}(x - a)^{\mu + \beta - 1}$$
(7)

and

$$(D_{a + }^{\alpha } f)(x) = \left( {\frac{d}{dx}} \right)^{n} (I_{a + }^{n - \alpha } f)(x).$$
(8)

Khan and Ahmed (2013) proved the following result.

If $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$; $$q \le Re(\alpha ) + p$$ and

$$\text{Re} \left( \alpha \right) > 0,\text{Re} \left( \beta \right) > 0,\text{Re} \left( \gamma \right) > 0,\text{Re} \left( \delta \right) > 0,\text{Re} \left( \mu \right) > 0,\text{Re} \left( \upsilon \right) > 0,\text{Re} \left( \rho \right) > 0,\text{Re} \left( \sigma \right) > 0$$

then for $$m \in N,$$

$$\left( {\frac{d}{dz}} \right)^{m} \left[ {z^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (wz^{\alpha } )} \right] = z^{\beta - m - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (wz^{\alpha } );\,\,\,\,\text{Re} \left( {\beta - m} \right) > 0.$$
(9)

In continuation of study, in this paper we give the operator associated with $$E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (z)$$ as follows:

Let $$f(x) \in L(a,b)$$, define

$$\left( {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f} \right)(x) = \int\limits_{a}^{x} {\left( {x - t} \right)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \left[ {w\left( {x - t} \right)^{\alpha } } \right]\,f\left( t \right)dt} ;\,\,\,\,x > a,$$
(10)

where $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon \in C$$; $$p,q,\rho ,\sigma > 0$$; $$q \le Re(\alpha ) + p$$; $$\rho \le \text{Re} \left( \sigma \right) + p$$; $$q \le \text{Re} \left( \sigma \right) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right)} \right) > 0.$$

## Main results

Using the definition (4), one can easily prove following lemma.

### Lemma 1

If $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$; $$q \le Re(\alpha ) + p$$; $$x > a$$; $$a \in R_{ + } = [0,\infty )$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,$$ then

$$E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (at^{\alpha } ) = \beta E_{\alpha ,\beta + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \left( {at^{\alpha } } \right) + \alpha tE_{\alpha ,\beta + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \left( {at^{\alpha } } \right)$$
(11)

### Theorem 1

Let $$a \in R_{ + } = [0,\infty )$$. Let $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$ ; $$q \le Re(\alpha ) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,x > a.$$ Then

$$\left( I_{a + }^{r} \left[(t - a)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(t - a)^{\alpha } \} \right)(x) = (x - a)^{r + \beta - 1} E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - a)^{\alpha } \right]$$
(12)
$$\left( D_{a + }^{r} \left[(t - a)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(t - a)^{\alpha } \} \right)(x) = (x - a)^{\beta - r - 1} E_{\alpha ,\beta - r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - a)^{\alpha } \right]$$
(13)

### Proof

Using definitions (3) and (5) and further simplification gives

\begin{aligned} & \left( I_{a + }^{r} \left[(t - a)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(t - a)^{\alpha } \} \right)(x)\right. \\ & {\kern 1pt} \quad = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{w^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} \left( {I_{a + }^{r} [(t - a)^{\alpha n + \beta - 1} ]} \right)(x) \\ & \quad = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{w^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} \frac{\Gamma (\alpha n + \beta )}{\Gamma (\alpha n + \beta + r)}(x - a)^{\alpha n + \beta + r - 1} \\ & \quad = (x - a)^{r + \beta - 1} \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta + r)}} \frac{{[w(x - a)^{\alpha } ]}}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}^{n} \\ & \quad =\left. (x - a)^{r + \beta - 1} E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - a)^{\alpha } \right]. \\ \end{aligned}

This completes the proof of (12).â–ˇ

To prove (13), we use definitions (8) and further simplification gives

\begin{aligned} & \left( D_{a + }^{r} \left[(t - a)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(t - a)^{\alpha } \} \right)(x)\right. \\ & \quad = \left( {\frac{d}{dx}} \right)^{n} \left( {I_{a + }^{n - r} [(t - a)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(t - a)^{\alpha } \} ]} \right)(x), \\ \end{aligned}

On applying (12) with replacement of $$r$$ by $$n - r$$, the above equation reduces to

$$= \left( {\frac{d}{dx}} \right)^{n} \left[(x - a)^{\beta + n - r - 1} E_{\alpha ,\beta + n - r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(x - a)^{\alpha } \} \right],$$

From (9), we get

\begin{aligned} &= (x - a)^{\beta + n - r - 1 - n} E_{\alpha ,\beta + n - r - n,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ w(x - a)^{\alpha } \} \hfill \\ &= (x - a)^{\beta - r - 1} E_{\alpha ,\beta - r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - a)^{\alpha } ]. \hfill \\ \end{aligned}

### Theorem 2

Let $$a \in R_{ + } = [0,\infty ),$$ $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$ and $$q \le Re(\alpha ) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0$$ $$x > a.$$ Then

$$\left( {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} (t - a)^{r - 1} } \right)(x) = (x - a)^{\beta + r - 1} \Gamma (r)E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \left( {w(x - a)^{\alpha } } \right).$$
(14)

### Proof

Taking $$f(t) = (t - a)^{r - 1}$$ in (10), we get

\begin{aligned} \left( {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} (t - a)^{r - 1} } \right)(x) & = \int\limits_{a}^{x} {(x - t)^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ](t - a)^{r - 1} dt} \\ & = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{w^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} \int\limits_{a}^{x} {(x - t)^{\alpha n + \beta - 1} (t - a)^{r - 1} dt} , \\ \end{aligned}

Replacing $$t$$ by $$a + (x - a)t$$ and simplifying the above equation

$$= \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{w^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} (x - a)^{\alpha n + \beta + r - 1} B(\alpha n + \beta ,r)$$

and further simplification of above equation gives the proof of Theorem 2.

### Theorem 3

Let $$a \in R_{ + } = [0,\infty ),$$ $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$ and $$q \le Re(\alpha ) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,$$ $$b > a.$$, Then the operator $$E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q}$$ is bounded on $$L(a,b)$$ and

$$\left\| {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f} \right\|_{1} \le B\left\| f \right\|_{1} ,$$
(15)

where

$$B = (b - a)^{{\text{Re} (\beta )}} \sum\limits_{n = 0}^{\infty } {\frac{{\left| {(\mu )_{\rho n} } \right|\left| {(\gamma )_{qn} } \right|}}{{[\text{Re} (\alpha )n + \text{Re} (\beta )]\left| {\Gamma (\alpha n + \beta )} \right|}}} \frac{{\left| {w(b - a)^{{\text{Re} (\alpha )}} } \right|^{n} }}{{\left| {(\upsilon )_{\sigma n} } \right|\left| {(\delta )_{pn} } \right|}}.$$
(16)

### Proof

On using the definition (10) and applying Dirichletâ€™s formula (Samko et al. 1993), we have

\begin{aligned} \left\| {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f} \right\|_{1} & = \int\limits_{a}^{b} {\left| {\int\limits_{a}^{x} {(x - t)^{\beta - 1} } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ]\,f(t)dt} \right|} dx \\ & \le \int\limits_{a}^{b} {\left[ {\int\limits_{t}^{b} {(x - t)^{{\text{Re} (\beta ) - 1}} } \left| {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ]} \right|dx} \right]\left| {f(t)} \right|} dt, \\ \end{aligned}

Taking $$u = x - t$$ in inner integral, thisÂ yields

\begin{aligned} \left\| {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f} \right\|_{1} & \le \int\limits_{a}^{b} {\left[ {\int\limits_{0}^{b - t} {u^{{\text{Re} (\beta ) - 1}} } \left| {E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [wu^{\alpha } ]} \right|du} \right]\left| {f(t)} \right|} dt \\ & \le \int\limits_{a}^{b} {\sum\limits_{n = 0}^{\infty } {\frac{{\left| {(\mu )_{\rho n} } \right|\left| {(\gamma )_{qn} } \right|}}{{\left| {\Gamma (\alpha n + \beta )} \right|}}\frac{{\left| w \right|^{n} }}{{\left| {(\upsilon )_{\sigma n} } \right|\left| {(\delta )_{pn} } \right|}}\left[ {\frac{{u^{{\text{Re} (\alpha )n + \text{Re} (\beta )}} }}{{\text{Re} (\alpha )n + \text{Re} (\beta )}}} \right]_{0}^{b - a} } \left| {f(t)} \right|} dt \\ & = (b - a)^{{\text{Re} (\beta )}} \sum\limits_{n = 0}^{\infty } {\frac{{\left| {(\mu )_{\rho n} } \right|\left| {(\gamma )_{qn} } \right|}}{{\left| {\Gamma (\alpha n + \beta )} \right|}}\frac{{\left| w \right|^{n} }}{{\left| {(\upsilon )_{\sigma n} } \right|\left| {(\delta )_{pn} } \right|}}\frac{{(b - a)^{{\text{Re} (\alpha )n}} }}{{[\text{Re} (\alpha )n + \text{Re} (\beta )]}}} \int\limits_{a}^{b} {\left| {f(t)} \right|dt} \\ \end{aligned}

This completes the proof.â–ˇ

### Theorem 4

(Composition with Riemannâ€“Liouville fractional integration operator) Let $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$; $$q \le Re(\alpha ) + p$$; $$b > a$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0.$$ Then the relation

$$I_{a + }^{r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f \equiv E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f \equiv E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} I_{a + }^{r} f$$
(17)

holds for any summable function $$f \in L(a,b).$$

### Proof

From (10) and (5), we get

$$(I_{a + }^{r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f)(x) = \frac{1}{\Gamma (r)}\int\limits_{a}^{x} {\int\limits_{a}^{u} {(x - u)^{r - 1} (u - t)^{\beta - 1} } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(u - t)^{\alpha } ]f(t)dt} du$$

Applying Dirichletâ€™s formula (Samko et al. 1993), we get

$$= \int\limits_{a}^{x} {\left[ {\frac{1}{\Gamma (r)}\int\limits_{t}^{x} {(x - u)^{r - 1} (u - t)^{\beta - 1} } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(u - t)^{\alpha } ]du} \right]} f(t)dt$$

Substituting $$u - t = \tau$$ in the above equation, we get

$$= \int\limits_{a}^{x} {\left[ {\frac{1}{\Gamma (r)}\int\limits_{0}^{x - t} {(x - t - \tau )^{r - 1} \tau^{\beta - 1} } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w\tau^{\alpha } ]d\tau } \right]} f(t)dt$$

Again using (5), this equation becomes

$$= \int\limits_{a}^{x} {\left( {I_{0 + }^{\mu } \left[ {\tau^{\beta - 1} E_{\alpha ,\beta }^{\gamma ,q} (w\tau^{\alpha } )\,} \right]} \right)} \,(x - t)\,f(t)dt,$$

Applying (12), this yields

$$= \int\limits_{a}^{x} {(x - t)^{\beta + r - 1} E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ]\,f(t)dt} ,$$

Using (10), we get

$$= (E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f)(x).$$

The other equality canÂ also be proved in the similar way.

### Theorem 5

Let $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$; $$q \le Re(\alpha ) + p$$; $$b > a$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0.$$ Then the relation

$$D_{a + }^{r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f \equiv E_{\alpha ,\beta - r,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f$$
(18)

holds for any continuous function $$f \in C[a,b]$$.

### Proof

From (8), we have

$$\left( D_{a + }^{r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f \right)(x) = \left( {\frac{d}{dx}} \right)^{n} \left( {I_{a + }^{n - r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f} \right)(x)$$

Again using Theorem 4 and definition (10),

$$= \left( {\frac{d}{dx}} \right)^{n} \left[ {\int\limits_{a}^{x} {(x - t)^{\beta + n - r - 1} } E_{\alpha ,\beta + n - r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ]f(t)dt} \right]$$
(19)

The integrand in the above equation is continuous function on $$[a,b]$$, here we take

$$\frac{d}{dx}\int\limits_{a}^{x} {h(x,t)dt = \int\limits_{a}^{x} {\frac{\partial }{\partial x}h(x,t)dt + h(x,x)} }$$
(20)

From (19) and (20), we get

$$\left( D_{a + }^{r} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p;w;a + }^{\mu ,\rho ,\gamma ,q} f \right)(x) = \left( {\frac{d}{dx}} \right)^{n - 1} \int\limits_{a}^{x} {(x - t)^{\beta + n - r - 2} } E_{\alpha ,\beta + n - r - 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [w(x - t)^{\alpha } ]f(t)dt$$

Applying same procedures as above, this led the proof of the theorem. This is easy to prove by using mathematical induction method also.

### Theorem 6

Let $$a \in R_{ + } = [0,\infty ),$$ $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0;$$ $$q \le Re(\alpha ) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,\,x > a.$$ Then

$$\left( I_{0 + }^{r} \left[t^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{\alpha } \} \right)(x) = x^{r + \beta - 1} E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [ax^{\alpha } ] \right.$$
(21)

### Proof

We have

\begin{aligned} \left( I_{0 + }^{r} \left[t^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{\alpha } \} \right. \right)(x) & = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{a^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} \left( {I_{0 + }^{r} [t^{\alpha n + \beta - 1} ]} \right)(x) \\ & = \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta )}\frac{{a^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }}} \frac{\Gamma (\alpha n + \beta )}{\Gamma (\alpha n + \beta + r)}(x)^{\alpha n + \beta + r - 1} \\ & = x^{r + \beta - 1} \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} }}{\Gamma (\alpha n + \beta + \mu )}} \frac{{[ax^{\alpha } ]^{n} }}{{(\upsilon )_{\sigma n} (\delta )_{pn} }} \\ & = x^{r + \beta - 1} E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [ax^{\alpha } ]. \\ \end{aligned}

This completes the proof.â–ˇ

### Corollary 1

If $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$ and $$q \le Re(\alpha ) + p$$ $$x > a$$; $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,\,a \in R_{ + } = [0,\infty ).$$ Let $$I_{0 + }^{r}$$ be the left-sided operator of Riemannâ€“Liouville fractional integral. Then

$$\left( I_{0 + }^{r} \left[ t^{\beta - 1} E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{\alpha } \} \right)(x)\right. = x^{r + \beta - 1} \{ (\beta + r)E_{\alpha ,\beta + r + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} + x\frac{d}{dx}E_{\alpha ,\beta + r + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (ax^{\alpha } )\}$$
(22)

Proof is very obvious from Lemma 1 and Theorem 6.

### Theorem 7

Let $$a \in R_{ + } = [0,\infty )$$, $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0;$$ $$q \le Re(\alpha ) + p$$ and $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,\,\,\,x > a,$$ $$I_{ - }^{r}$$ be the right-sided operator of Riemannâ€“Liouville fractional integral. Then

$$\left( {I_{ - }^{r} [t^{ - r - \beta } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{ - \alpha } \} } \right)(x) = x^{ - \beta } E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} [ax^{ - \alpha } ]$$

### Proof

Let

$$\left( {I_{ - }^{r} [t^{ - r - \beta } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{ - \alpha } \} } \right)(x) = \frac{1}{\Gamma (r)}\int\limits_{x}^{\infty } {(t - x)^{r - 1} t^{ - r - \beta } \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} (at^{ - \alpha } )^{n} }}{{\Gamma (\alpha n + \beta )(\upsilon )_{\sigma n} (\delta )_{pn} }}} dt}$$

On changing the order of the summation and integration then afterward applying beta function, this gives

$$= x^{ - \beta } \sum\limits_{n = 0}^{\infty } {\frac{{(\mu )_{\rho n} (\gamma )_{qn} (ax^{ - \alpha } )^{n} }}{{\Gamma (\alpha n + \beta + r)(\upsilon )_{\sigma n} (\delta )_{pn} }}} = x^{ - \beta } E_{\alpha ,\beta + r,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ ax^{ - \alpha } \}$$

### Corollary 2

If $$\alpha ,\beta ,\gamma ,\delta ,\mu ,\upsilon ,\rho ,\sigma \in C$$; $$p,q > 0$$ and $$q \le Re(\alpha ) + p; \,\,\, x > a$$ ; $$\hbox{min} \left( {\text{Re} \left( \alpha \right),\text{Re} \left( \beta \right),\text{Re} \left( \gamma \right),\text{Re} \left( \delta \right),\text{Re} \left( \mu \right),\text{Re} \left( \upsilon \right),\text{Re} \left( \rho \right),\text{Re} \left( \sigma \right)} \right) > 0,$$ $$a \in R_{ + } = [0,\infty )$$. Let $$I_{ - }^{r}$$ be the right-sided operator of Riemannâ€“Liouville fractional integral. Then

$$\left( {I_{ - }^{r} [t^{ - r - \beta } E_{\alpha ,\beta ,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} \{ at^{ - \alpha } \} } \right)(x) = x^{ - r - \beta } \{ (\beta + r)E_{\alpha ,\beta + r + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} + x\frac{d}{dx}E_{\alpha ,\beta + r + 1,\upsilon ,\sigma ,\delta ,p}^{\mu ,\rho ,\gamma ,q} (ax^{ - \alpha } )\}$$
(23)

## Conclusion

In this paper, we proved some properties of generalized Mittag-Leffler functions and also used the fractional calculus approach to prove Theorems 4, 5, 6 and 7.

## References

• Khan MA, Ahmed S (2013) On some properties of the generalized Mittag-Leffler function. SpringerPlus 2:1â€“9

• Kilbas AA, Sagio M, Saxena RK (2004) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Trans Spec Funct 15(1):31â€“49

• Miller KS, Ross B (1993) An introduction to fractional calculus and fractional differential equations. Wiley, New York

• Mittag- Leffler G (1903) Sur la nouvelle function E Î±(x). C R Acad Sci Paris 137:554â€“558

• Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, theory and applications. Gordon and Breach, New York

• Shukla AK, Prajapati JC (2007) On a generalization of Mittag-Leffler function and its properties. J Math Anal Appl 336(2):797â€“811

## Authorsâ€™ contributions

The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

## Author information

Authors

### Corresponding author

Correspondence to A. K. Shukla.

## Rights and permissions

Reprints and permissions

Desai, R., Salehbhai, I.A. & Shukla, A.K. Note on generalized Mittag-Leffler function. SpringerPlus 5, 683 (2016). https://doi.org/10.1186/s40064-016-2299-x