A Study on Generalized Multivariable Mittag-Leffler Function via Generalized Fractional Calculus Operators

Abstract

We study some properties of generalized multivariable Mittag-Leffler function. Also we establish two theorems, which give the images of this function under the generalized fractional integral operators involving Fox’s H-function as kernel. Relating affirmations in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some known special cases have also been mentioned in the concluding section.

1. Introduction and Preliminaries

Recently, Gurjar et al. [1] introduced a multivariable generalized Mittag-Leffler (M-L) function; this function and its special cases have recently found various essential applications in solving problems in physics, biology, engineering and applied sciences (see; [2–5]). The function is defined for and aswhereSome important special cases of the multivariable generalized M-L function are enumerated below:(1)If in (1), the function reduces to multivariate analogue of generalized M-L function which was defined by Saxena et al. [6] aswhere and (2)If we set in (1), then the function reduces to another type of multivariate generalized M-L function which was also defined by Saxena et al. [6] aswhere and (3)If we put , equation (1) reduces to generalized M-L function which was defined by Salim and Faraj [7] aswhere (4)By setting , and in (1) considered by Shukla and Prajapati [8], in addition to that, if , defined by Prabhakar [9].(5)If and in Eq. (1), it reduces to Wiman’s function [10], moreover if , Mittag-Leffler function [11] will be the result.

In the present paper, our aim is to study some fundamental properties of multivariable generalized M-L function defined in equation (1). For that, we consider two generalized fractional integral operators involving Fox’s H-function as kernel, defined by Kalla [12, 13] and further studied by Srivastava and Buschman [14]. Recently, Garg, Rao and Kalla [15] studied some fractional calculus properties of M-L type function, involving these fractional calculus operators. We use the following notations for the left-sided and right-sided generalized fractional integral operators:andwhere and represent the expressions and respectively, with . Here the symbol stands for well-known Fox’s H-function, defined by means of the following Mellin-Barnes type integral [16]:whereand is a suitable contour in . The orders are integers, and the parameters are such that For the conditions of analyticity of the H-function and other details, one can see [16, 17]. Throughout the present paper, we assume that these conditions are satisfied by the H-function.

For our purpose, we recall the definition of generalized Wright hypergeometric function (see, for details, Srivastava and Karlsson [18]), for and , with defined as follows:The generalized Wright function was introduced by Wright [19] in the form of (10) under the condition:

2. Images of the Generalized Multivariable M-L Function under 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 such that and be the generalized left-side fractional integral operator (6), then there hold the result true:provided that(i)(ii)(iii)(iv), where

Proof. Using the definition (6) in the left hand side of (12), writing the functions in the form given by (1) and (8), interchanging the order of integration and summations under the statement of Theorem 1, we obtainTo evaluate the -integral substituting , we obtainFinally on evaluating the integral as beta integral and re-interpreting the result in terms of H-Function and generalized multivariable M-L function, we easily arrive at the result (12).

Theorem 2. Assume such that and be the generalized right-side fractional integral operator (7), thenprovided that(i)(ii)(iii)(iv) where

Proof. Proceeding as in Theorem 1, one can easily prove the Theorem 2. Therefore, we omit the detailed proof of Theorem 2.

3. Images of the Generalized Multivariable M-L Function under the Saigo’s Fractional Integral Operators

If we choose and in the Theorems 1 and 2, then the fractional integral operators and reduce to the corresponding Saigo’s operators [20, 21]. These operators are connected by the following functional relations:andwhere the operators and respectively denote the Saigo’s left-side and right-side fractional integral operators and are defined as and

Lemma 3. Let Then their exists the relation (1)If and , then(2)If and , then

Now, on using the above mentioned substitutions and relations, we obtain the following images of the generalized multivariable M-L function under the fractional integral operators of Saigo type:

Corollary 4. Let ; such that , then the following formula holds:

Proof. Denote L.H.S. of Corollary 4 by . By virtue of (1) and (18), we havewhich upon Lemma 3(1), yieldsUsing the definition of (10) in the right-hand side of (24), we arrive at the result (22).

Corollary 5. Assume such that , then

Proof. By a similar manner as in proof of Corollary 4 by using Lemma 3(2), we get the desired formula (25).

4. Images of the Generalized Multivariable Mittage-Leffler Function under the Erdélyi-Kober Fractional Integral Operators

If we take in the Saigo’s fractional integral operators (18) and (19), then due to Saigo [20], we getandwhere the Erdélyi-Kober fractional integral operators are defined byandWe now give images of the generalized multivariable M-L function under the Erdélyi-Kober fractional integral operators:

Corollary 6. Let such that , then

Corollary 7. Let such that , then

5. Special Cases

In this section, we consider some consequences and applications of the results derived in the previous sections. If we take , then the fractional calculus operators (18) and (19), respectively, reduce to the Riemann-Liouville and Weyl fractional integral operators. Hence, we obtain the following image formulas:

Corollary 8. Let such that , then the following formula holds:where the Riemann-Liouville fractional integral operator is defined by

Corollary 9. Assume such that , then the following result is true:where the Weyl type fractional integral operator is given by

6. Concluding Remark

The generalized multivariable M-L function is interesting due to the various (described in introduction section) M-L functions that follow as its particular cases, so the generalized fractional calculus formulas are deduced in this communication, and we can find many applications giving the Saigo, Erdelyi-Kober, Riemann-Liouville and Weyl type fractional integrals of aforementioned functions on taking special cases into account. For various other special cases we refer to [22–27] and we left results for the interested readers.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.