# Generalized Fractional Integral Operators Involving Mittag-Leffler Function.

1. Introduction and Preliminaries

M-L Function. In 1903, Mittag-Leffler  introduced the function [E.sub.[lambda]] (z), defined by

[mathematical expression not reproducible] (1)

A further, two-index generalization of this function was given by Wiman  as

[mathematical expression not reproducible] (2)

where R([lambda]) > 0 and R([beta]) > 0.

By means of the series representation a generalization of M-L function (2) is introduced by Prabhakar  as

[mathematical expression not reproducible], (3)

where [lambda], [beta], [gamma] [member of] C (R([lambda]) > 0). Further, it is an entire function of order [[R([lambda])].sup.-1].

Generalized Fractional Integral Operator. Now, we recall the definition of generalized fractional integral operators involving Fox's H-function as kernel, defined by Saxena and Kumbhat  means of the following equations:

[mathematical expression not reproducible] (4)

[mathematical expression not reproducible], (5)

where U and V represent the expressions

[mathematical expression not reproducible], (6)

respectively, with [tau], v > 0. The sufficient conditions of operators are given below:

(i) 1 [less than or equal to] p, q < [infinity], [p.sup.-1] +[q.sup.-1] - 1;

(ii) [mathematical expression not reproducible];

(iii) f(x) [member of] [L.sub.p](0, [infinity]);

(iv) [mathematical expression not reproducible].

An interest in the study of the fractional calculus associated with the Mittag-Leffler function and H-function, its application in the form of differential, and integral equations of, in particular, fractional orders (see [5-10]).

H-Function. Symbol [H.sup.m,n.sub.p,q](x) stands for well known Fox H-function , in operator (4) and (5) defined in terms of Mellin-Barnes type contour integral as follows:

[mathematical expression not reproducible], (7)

where

[mathematical expression not reproducible], (8)

[mathematical expression not reproducible].

For the conditions of analytically continuations together with the convergence conditions of H-function, one can see [12,13]. Throughout the present paper, we assume that these conditions are satisfied by the function.

2. Images of M-L Function Involving the Generalized Fractional Integral Operators

In this section, we consider two generalized fractional integral operators involving the Fox's H-function as the kernels and derived the following theorems.

Theorem 1. Let [mathematical expression not reproducible]; then the fractional integration [R.sup.[mu],[alpha].sub.x,r] of the product of M-L function exists, under the condition

[mathematical expression not reproducible]; (9)

then there holds the following formula:

[mathematical expression not reproducible]. (10)

Proof. Let l be the left-hand side of (10); using (3) and (4), we have

[mathematical expression not reproducible]. (11)

Changing the order of the integration valid under the condition given with the theorem, we obtain

[mathematical expression not reproducible]. (12)

Let the substitution [t.sup.r]/[x.sup.r] = w; then t = x[w.sup.(1/r)] in the above term; we get

[mathematical expression not reproducible]. (13)

Using beta function for (13), the inner integral reduces to

[mathematical expression not reproducible]. (14)

Interpreting the right-hand side of (14), in view of the definition (7), we arrive at the result (10).

Theorem 2. Let [mathematical expression not reproducible]; then the fractional integration [K.sup.[epsilon],[alpha].sub.x,r] of the product of M-L function exists, under the condition

[mathematical expression not reproducible] (15)

and then the following formula holds:

[mathematical expression not reproducible]. (16)

Proof. Let p be the left-hand side of (16); using (3) and (5), we have

[mathematical expression not reproducible]. (17)

Changing the order of the integration valid under the condition given with the theorem statement, we obtain

[mathematical expression not reproducible]. (18)

Letting the substitution [x.sup.r]/[t.sup.r] = u, then t = x/[u.sup.(1/r)] in the above term and, using beta function, we get

[mathematical expression not reproducible]. (19)

Interpreting the right-hand side of (19), in view of definition (7), we arrive at the result (16).

3. Integral Transforms of Fractional Integral Involving M-L Function

In this section, Mellin, Laplace, Euler, Whittaker, and K-transforms of the results established in Theorems 1 and 2 have been obtained.

Euler Transform (Sneddon ). The Euler transform of a function f(t) is defined as

[mathematical expression not reproducible]. (20)

Theorem 3. Let [mathematical expression not reproducible]; then

[mathematical expression not reproducible]. (21)

Proof. Using (10) and (20) gives

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible]. (23)

Now, we obtain the result (23). This completes the proof of the theorem.

Theorem 4. Let [mathematical expression not reproducible]; then

[mathematical expression not reproducible]. (24)

Proof. In similar manner, in proof of Theorem 3, we obtain the result (24).

Mellin Transform (Debnath and Bhatta ). The Mellin transform of a function f(t) is defined as

[mathematical expression not reproducible]. (25)

Theorem 5. All conditions follow from that stated in Theorem 1 with R(s) > R(v); the following result holds:

[mathematical expression not reproducible]. (26)

Proof. From (10) and (25), it gives

[mathematical expression not reproducible]. (27)

Now, evaluating the Mellin transform of [t.sup.[??]+vn-1] using formula given by Mathai et al. . we arrive at (26).

Theorem 6. All conditions follow from what is stated in Theorem 2 with R(1 - [??]) < 1, R(s) > R(v); the following result holds:

[mathematical expression not reproducible]. (28)

Proof. In similar manner, in proof of Theorem 5, we obtain the result (28).

Laplace Transform (Sneddon ). The Laplace transform of a function /(i), denoted by F(s), is defined by the equation

[mathematical expression not reproducible]. (29)

Provided the integral (29) is convergent and that the function, f(t), is continuous for t > 0 and of exponential order as t [right arrow] [infinity], (29) maybe symbolicallywritten as

F(s) = L{f(t); s} or f(t) - [L.sup.-1] {F(s);i}. (30)

The following result is well known:

[mathematical expression not reproducible]. (31)

Theorem 7. All conditions follow from what is stated in Theorem 1 with R(s) > 0 and R([??] + vn) > 0; the following result holds:

[mathematical expression not reproducible]. (32)

Proof. we can develop similar line by using result of Laplace integral (31).

Theorem 8. All conditions follow from what is stated in Theorem 2 with R(s) > 0 and R(1 - [??] - vn) > 0; the following result holds:

[mathematical expression not reproducible]. (33)

Proof. In a similar manner, in proof of Theorem 7, we obtain the result (33).

Whittaker Transform (Whittaker and Watson ). Due to Whittaker transform, the following result holds:

[mathematical expression not reproducible], (34)

where R([omega][+ or -][zeta]) > -1/2 and [W.sub.[chi],[omega]](t) is the Whittaker confluent hypergeometric function:

[mathematical expression not reproducible], (35)

where [M.sub.[chi],[omega]] is defined by

[mathematical expression not reproducible]. (36)

Theorem 9. Following what is stated in Theorem 1for conditions on parameters, with R[[omega][+ or -]([??] + [zeta] + vn - 1)] > 1/2, then the following result holds:

[mathematical expression not reproducible]. (37)

Proof. Using (10) and (34), it gives

[mathematical expression not reproducible]. (38)

Assume that t = k, [??] dt = dk/[phi]; we get

[mathematical expression not reproducible]. (39)

Interpreting the right-hand side of (39), using (34), we arrive at the result (37).

Theorem 10. Following what is stated in Theorem 2 for conditions on parameters, with R[[omega][+ or -](-[??] + [zeta]- vn - 1)] > 1/2,then the following result holds:

[mathematical expression not reproducible]. (40)

Proof. In a similar manner, in proof of Theorem 9, we obtain the result (40).

K-Transform (Erdelyi et al. ). This transform is defined by the following integral equation:

[mathematical expression not reproducible], (41)

where R(p) > 0; [K.sub.v](x) is the Bessel function of the second kind defined by (, p. 332)

[mathematical expression not reproducible], (42)

where [W.sub.0,v](x) is the Whittaker function defined in Erdelyi et al. .

The following result given in Mathai et al. (, p. 54, eq. 2.37) will be used in evaluating the integrals:

[mathematical expression not reproducible]. (43)

Theorem 11. Following what is stated in Theorem 1for conditions on parameters, with R([omega]) > 0; R(([rho] + [??] + vn - 1)[+ or -]l) > 0, then the following result holds:

[mathematical expression not reproducible]. (44)

Proof. Using (10) and (44), it gives

[mathematical expression not reproducible], (45)

and we get

[mathematical expression not reproducible]. (46)

Interpreting the right-hand side of (46), we arrive at the result (44).

Theorem 12. Following what is stated in Theorem 2 for conditions on parameters, with R([omega]) > 0; R(([rho] - [??] - vn) [+ or -]l) > 0, then the following result holds:

[mathematical expression not reproducible]. (47)

Proof. In a similar manner, in proof of Theorem 11, we obtain the result (47).

4. Properties of Integral Operators

Here, we established some properties of the operators as consequences of Theorems 1 and 2. These properties show compositions of power function.

Theorem 13. Following all the conditions on parameters as stated in Theorem 1 with R([psi] + [??]) > 0, then the following result holds true:

[mathematical expression not reproducible]. (48)

Proof. From (10), the left-hand side of (48), we have

[mathematical expression not reproducible], (49)

and again, by (10), the right-hand side of (48) follows:

[mathematical expression not reproducible]. (50)

It seems that Theorem 13 readily follows due to (49) and (50).

Theorem 14. Following all the conditions on parameters as stated in Theorem 2 with R([beta] + [??]) > 0, then the following result holds true:

[mathematical expression not reproducible]. (51)

Proof. From (12), the left-hand side of (51), we have

[mathematical expression not reproducible]. (52)

Again by (12), the right-hand side of (51) follows:

[mathematical expression not reproducible]. (53)

It seems that Theorem 14 readily follows due to (52) and (53).

5. Conclusions

In this article, we have investigated and studied two classes of generalized fractional integral operators involving Fox's H-function as kernel due to Saxena and Kumbhat which are applied on M-L function. We discussed the actions of fractional integral operators under Euler, Mellin, Laplace, Whittaker, and K-transforms and results are given in better pragmatic series solutions. The majority of the results derived here are general in nature and compact forms are fairly helpful in deriving a variety of integral formulas in the theory of integral operators which arises in a range of problems of applied sciences like kinematics, diffusion equation, kinetic equation, fractal geometry, anomalous diffusion, propagation of seismic waves, turbulence, etc. We may obtain other special functions such as M-L function and Bessel-Maitland function (see, e.g., ([19-21]) as its special cases and, therefore, various unified fractional integral presentations can be obtained as special cases of our results.

https://doi.org/10.1155/2018/7034124

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

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Hafte Amsalu (iD) and D. L. Suthar (iD)

Department of Mathematics, Wollo University, P.O. Box 1145, Dessie, Ethiopia

Correspondence should be addressed to Hafte Amsalu; yohanahafte@gmail.com

Received 2 March 2018; Accepted 11 April 2018; Published 3 June 2018

Academic Editor: Khalil Ezzinbi
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