Euler type integral operator involving k-Mittag-Leffler function

TL;DR

This paper explores a Euler type integral operator that incorporates the k-Mittag-Leffler function, extending previous definitions and examining special cases to deepen understanding of its properties.

Contribution

It introduces a new Euler type integral operator involving the k-Mittag-Leffler function and analyzes its special cases, expanding the theoretical framework.

Findings

Derived new integral operator involving k-Mittag-Leffler function

Analyzed special cases of the operator

Extended existing mathematical definitions

Abstract

This paper deals with a Euler type integral operator involving k-Mittag-Leffler function defined by Gupta and Parihar [8]. Furthermore, some special cases are also taken into consideration.

Equations

$${}_{p}\Psi_{q}^{k}\left[\begin{array}[]{lll}(\alpha_{1},A_{1}),...,(\alpha_{p},A_{p});\\ &z\\ (\beta_{1},B_{1}),...,(\beta_{p},B_{p});\end{array}\right]={}_{p}\Psi_{q}^{k}\left((\alpha_{j},A_{j})_{1,p};(\beta_{j},B_{j})_{1,q};z\right)$$

$$\underset{0}{\overset{1}{\int }}{t}^{\frac{a}{k}-1}(1-t{)}^{\frac{b}{k}-1}\exp \left(\frac{-A}{t^{\frac{m}{k}}(1-t{)}^{\frac{m}{k}}}\right)E_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z{t}^{\frac{\alpha }{k}})dt$$

$$I_1=\underset{0}{\overset{1}{\int }}{t}^{\frac{a}{k}-1}(1-t{)}^{\frac{b}{k}-1}\exp \left(\frac{-A}{t^{\frac{m}{k}}(1-t{)}^{\frac{m}{k}}}\right)\sum _{n=0}^{\infty }\frac{(\gamma {)}_{qn,k}{z}^n{t}^{\frac{n\alpha }{k}}}{{\Gamma }_k(\alpha n+\beta )(\delta {)}_{pn,k}}dt$$

$$I_1=\sum _{n=0}^{\infty }\frac{(\gamma {)}_{qn,k}{z}^n}{{\Gamma }_k(\alpha n+\beta )(\delta {)}_{pn,k}}\underset{0}{\overset{1}{\int }}{t}^{\frac{a+n\alpha }{k}-1}(1-t{)}^{\frac{b}{k}-1}\exp \left(\frac{-A}{t^{\frac{m}{k}}(1-t{)}^{\frac{m}{k}}}\right)dt$$

$$\frac{1}{{\Gamma }_k(b)}\underset{0}{\overset{1}{\int }}{t}^{\frac{\beta }{k}-1}(1-t{)}^{\frac{b}{k}-1}{E}_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z{t}^{\frac{\alpha }{k}})dt$$

$$\underset{t}{\overset{x}{\int }}(x-t{)}^{\frac{\rho }{k}-1}(s-t{)}^{\frac{\mu }{k}-1}\exp \left(\frac{-A}{(x-s{)}^{\frac{m}{k}}(s-t{)}^{\frac{m}{k}}}\right)E_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z(s-t{)}^{\frac{\alpha }{k}})dt$$

$$\underset{t}{\overset{x}{\int }}(x-t{)}^{\frac{\rho }{k}-1}(s-t{)}^{\frac{\mu }{k}-1}\exp \left(\frac{-A}{(x-s{)}^{\frac{m}{k}}(s-t{)}^{\frac{m}{k}}}\right)E_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z(s-t{)}^{\frac{\alpha }{k}})dt,$$

$$=\underset{0}{\overset{1}{\int }}(1-u{)}^{\frac{\rho }{k}-1}(x-t{)}^{\frac{\rho }{k}-1}{u}^{\frac{\mu }{k}-1}(x-t{)}^{\frac{\mu }{k}-1}\exp \left(\frac{-A}{((x-t)(1-u){)}^{\frac{m}{k}}(u(x-t){)}^{\frac{m}{k}}}\right)E_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z{u}^{\frac{\alpha }{k}}(x-t{)}^{\frac{\alpha }{k}})dt.$$

$$\underset{0}{\overset{1}{\int }}(1-u{)}^{\frac{\rho }{k}-1}(x-t{)}^{\frac{\rho }{k}-1}{u}^{\frac{\mu }{k}-1}(x-t{)}^{\frac{\mu }{k}-1}\sum _{r=0}^{\infty }\frac{(-A{)}^r}{(x-t{)}^{\frac{2mr}{k}}(1-u{)}^{\frac{mr}{k}}{u}^{\frac{mr}{k}}r!}\sum _{n=0}^{\infty }\frac{(\gamma {)}_{qn,k}{z}^n{u}^{\frac{n\alpha }{k}}(x-t{)}^{\frac{\alpha n}{k}}}{{\Gamma }_k(\alpha n+\beta )(\delta {)}_{pn,k}}.$$

$$=\sum _{r=0}^{\infty }\sum _{n=0}^{\infty }\frac{(x-t{)}^{\frac{\rho +\mu +n\alpha -2mr}{k}-2}(-A{)}^r(\gamma {)}_{qn,k}{z}^n}{{\Gamma }_k(\alpha n+\beta )(\delta {)}_{pn,k}r!}\underset{0}{\overset{1}{\int }}(1-u{)}^{\frac{\rho -mr}{k}-1}{u}^{\frac{\mu +n\alpha -mr}{k}-1}dt,$$

$$\underset{0}{\overset{1}{\int }}{t}^{\frac{\lambda }{k}-1}(1-t{)}^{\frac{\mu -\lambda }{k}-1}{\left(1-u{t}^{\frac{\rho }{k}}(1-t{)}^{\frac{\sigma }{k}}\right)}^{-a}\exp \left(\frac{-A}{t^{\frac{m}{k}}(1-t{)}^{\frac{m}{k}}}\right)E_{k,\alpha ,\beta ,p}^{\gamma ,\delta ,q}(z{t}^{\frac{\alpha }{k}})dt$$

Full text

**Euler type integral operator involving k-Mittag-Leffler function **

Waseem A. Khan1, Nisar K S2,∗ and Moin Ahmad3

1,3Department of Mathematics, Faculty of Science, Integral University, Lucknow-226026, India

*2,∗*Department of Mathematics, College of Arts and Science at Wadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University, Alkharj, Kingdom of Saudi Arabia

E-mail: waseem08_khan@rediffmail.com, n.sooppy@psau.edu.sa, moinah1986@gmail.com

Abstract. This paper deals with a Euler type integral operator involving k-Mittag-Leffler function defined by Gupta and Parihar [8]. Furthermore, some special cases are also taken into consideration.

Keywords: Euler type integrals, extended k-beta function, generalized k-Mittag-Leffler function, generalized k-Wright function.

2010 Mathematics Subject Classification.: 33E12, 33C45.

**1. Introduction **

Many authors namely, Diaz et al. [6], Kokologiannaki [11], Krasniqi [12], Mansour [17], Merovci [15], had introduced k-generalized gamma, k-Zeta, k-Beta functions. They had proven a number of their properties and inequalities for the above k-generalization functions. They had studied k-hypergeometric functions based on k-Pochhamer symbols for factorial functions. Propose our present research, we begin by mentioning the following definitions of some well known functions:

The integral representation of the k-gamma function as:

[TABLE]

and k-beta function is defined as:

[TABLE]

The generalized k-Wright function [7] represented as follows:

[TABLE]

[TABLE]

In 1903, the Swedish Mathematician introduced the Mittag-Leffler function $E_{\alpha}(z)$ [16] as:

[TABLE]

where $z\in\mathbb{C}$ and $\Gamma(s)$ is the Gamma function; ${\alpha}\geq 0$.

The Mittag-Leffler function is a direct generalization of $\exp(z)$ in which $\alpha=1$. Mittag-Leffler function naturally occurs as the solution of fractional order differential equation or fractional order integral equations.

A generalization of $E_{\alpha}(z)$ was studied by Wiman [26, 27] where he defined the function $E_{\alpha,\beta}(z)$ as:

[TABLE]

where $\alpha,\beta\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0.$ Which is also known as Mittag-Leffler function or Wiman’s function.

Prabhakar [18] introduced the function $E_{\alpha,\beta}^{\gamma}(z)$ in the form (see Killbas et al. [9]):

[TABLE]

where $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0.$

Shukla and Prajapati [24] (see Srivastava and Tomovaski [25]) defined and investigated the function $E_{\alpha,\beta}^{\gamma,q}(z)$ as:

[TABLE]

where $\alpha,\beta,\gamma\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0,q\in(0,1)\bigcup\mathbb{N}$ and $(\gamma)_{qn}=\dfrac{\Gamma(\gamma+qn)}{\Gamma(\gamma)}$ denotes the generalized Pochhammer symbol.

Salim [22] introduced a new generalized Mittag-Leffler function and defined it as:

[TABLE]

where $\alpha,\beta,\gamma,\delta\in\mathbb{C};\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0,\Re(\delta)>0.$

Afterward, Salim and Faraj [21] introduced the generalized Mittag-Leffler function$E_{\alpha,\beta,p}^{\gamma,\delta,q}(z)$ which is defined as:

[TABLE]

where ${\alpha,\beta,\gamma,\delta}\in\mathbb{C},\min\{\Re(\alpha),\Re(\beta),\Re(\gamma),\Re(\delta)\}>0;p,q>0$ and ${q<\Re\alpha+p}.$

Currently, Dorrego and Cerutti [5] introduced the generalized k-Mittag-Leffler function as follows:

[TABLE]

where ${k\in\mathbb{R};\alpha,\beta,\gamma,\delta}\in\mathbb{C},\min\{\Re(\alpha),\Re(\beta),\Re(\gamma),\Re(\delta)\}>0;p,q>0$ and ${q<\Re\alpha+p}.$

Now, we state the classical beta function denoted by $B(a,b)$ which is defined (see [14], see also [19]) by Euler’s integral as:

[TABLE]

In 1997, Chaudhary et al. [4] presented the following extension of Euler’s Beta function as follows:

[TABLE]

In continuation of his work, Lee et al. [13] introduced the generalizations of Euler beta functions and defined it as:

[TABLE]

where $\Re(p)>\Re(m)>0$.

In this paper, we consider the new generalizations of Euler type k-Beta functions as follows:

[TABLE]

where $k\in\mathbb{R};\Re(p)>\Re(m)>0$.

Clearly, when $m=k=1$, equation (1.14) reduces to (1.12) and further, by taking $p=0$ in (1.14), we get (1.11).

In this paper, we have obtained some theorems on Euler type integral operator involving generalized k-Mittag-Leffler function and have discussed some special cases.

**2. Basic properties of Euler type integral operator involving generalized k-Mittag-Leffler function **

Theorem 2.1. If ${{k\in\mathbb{R}},\alpha,\beta,\gamma,\delta}\in\mathbb{C},\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0,\Re(\delta)>0,\Re(A)>0$, $p,q>0$ and $q<R{\alpha}+p$, then,

[TABLE]

[TABLE]

Proof. In order to derive (2.1), we denote L.H.S. of (2.1) by $I_{1}$ and then expanding $E_{k,\alpha,\beta,p}^{\gamma,\delta,q}(zt^{\alpha})$ by using (1.10), to get:

[TABLE]

Now changing the order of summation and integration (which is guaranteed under the given conditions), we get:

[TABLE]

By using (1.14) as in the above equation, we attain the required result.

Corollary 2.1. For $A=0,a={\beta}$ in Theorem 2.1, we deduce the following result:

[TABLE]

[TABLE]

Theorem 2.2. If ${k\in\mathbb{R},\alpha,\beta,\gamma,\delta,\rho,\mu}\in\mathbb{C},\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0,\Re(\delta)>0,\Re(\rho),\Re(\mu),\Re(A)>0$, $p,q>0$ and $q<R{\alpha}+p$; $\mid arg(\frac{b^{{}^{\prime}}c+d}{a^{{}^{\prime}}c+d})\mid<{\pi}$, then

[TABLE]

[TABLE]

Proof. On taking L.H.S. of (2.2) and then by changing the variable s to $u=\frac{s-t}{x-t}$, we get:

[TABLE]

[TABLE]

Expanding the exponential function and k-Mittag-Leffler function in their respective series, we attain:

[TABLE]

By changing the order of summation and integration (which is guaranteed under the given conditions), we get:

[TABLE]

which further on using the integral(1.11) yields the required result.

Corollary 2.2. For $A=0,\mu=\beta$ in Theorem 2.2, we get:

[TABLE]

Theorem 2.3. If ${k\in\mathbb{R},\alpha,\beta,\gamma,\delta,\rho,\mu,\lambda,\sigma}\in\mathbb{C},\Re(\alpha)>0,\Re(\beta)>0,\Re(\gamma)>0,\Re(\sigma)>0,\Re(\delta)>0,\Re(\rho),\Re(\mu),\Re(\lambda)>0,\Re(A)>0$; $p,q>0$ and $q<\Re{\alpha}+p$, then

[TABLE]

[TABLE]

Proof. On taking L.H.S. of Theorem 2.3, using the definition of generalized k-Mittag-Leffler function (1.10), and then by changing the order of summation and integration, we get:

[TABLE]

[TABLE]

[TABLE]

By using (1.14) as in the above equation, we derive the required result.

Corollary 2.3. For $a=0$ in Theorem 2.3 reduces to the following result:

[TABLE]

[TABLE]

Corollary 2.4. Setting $a=0$, $A=0$ in Theorem 2.3, we deduces the following result:

[TABLE]

Remark. If we consider $p=q=k=m=1$ in Theorem (2.1), (2.2) and (2.3), we get a new class of Beta type integral operators involving the generalized Mittag-Leffler function defined by Salim [22] and the case $\delta=p=k=1$ of (2.1), (2.3) and (2.5) is seen to yield the known results of Ahmed and Khan [1].

**3. Special Cases **

In this section, we establish the following potentially useful integral operators involving generalized k-Beta type functions as special cases of our main results:

1. On setting $\gamma=q=1$ in Theorem 2.1, we get:

[TABLE]

[TABLE]

2. On setting $\alpha=\beta=q=\gamma=1$ in Theorem 2.1, we find:

[TABLE]

[TABLE]

3. On setting $\delta=p=1$ in Theorem 2.1, we attain:

[TABLE]

[TABLE]

4. On setting $\gamma=q=1$ in Theorem 2.2, we achieve:

[TABLE]

[TABLE]

5. On setting ${\alpha}={\beta}={\gamma}=q=1$ in Theorem 2.2, we acquire:

[TABLE]

[TABLE]

6. On setting ${\delta}=p=1$ in Theorem 2.2, we found:

[TABLE]

[TABLE]

7. On setting ${\gamma=q=1}$ in Theorem 2.3, we find:

[TABLE]

[TABLE]

8. On setting ${\alpha}={\beta}={\gamma}=q=1$ in Theorem 2.3, we get:

[TABLE]

[TABLE]

**References

**

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