Abstract
In this paper, we have studied a unified multi-index Mittag–Leffler function of several variables. An integral operator involving this Mittag–Leffler function is defined, and then, certain properties of the operator are established. The fractional differential equations involving the multi-index Mittag–Leffler function of several variables are also solved. Our results are very general, and these unify many known results. Some of the results are concluded at the end of the paper as special cases of our primary results.
1. Introduction
Recently, Mittag–Leffler (M-L) functions have demonstrated their special connection to fractional calculus, with a particular emphasis on fractional calculus problems arising from implementations. Several new special functions and implementations have been discovered over the last few decades. The advancement of research in the new era of special functions and their applications in mathematical modelling continues to attract many scientists from various disciplines (see recent papers; [1–13]).
The Mittag–Leffler function is extended to multi-index function in the following form [14, 15]:where ; and is an arbitrary complex number, i.e., .
If we make in (1) it reduces to the multi-index M-L function studied by Kiryakova [16, 17].
A multivariable extension of Mittag–Leffler function widely studied by Gautam [18], and also by Saxena et al. ([19], p. 547, Equation (7.1)), is defined and represented as follows:where ; and .
Motivated by the work on these functions, we consider here the subsequent multivariable and multi-index Mittag–Leffler function:where ; ;
We have also studied here, the integral operator involving the function defined by (3), as follows:with .
The Riemann–Liouville fractional derivative operator is defined as follows [20]:where is the fractional integral operator defined by
The elementary definitions are also required to be mentioned as follows.
The Laplace transform of fractional derivative is given as
Also, the formula for Laplace transform is
2. Results Required
The integral for the generalized M-L function defined in (3) is given by
The result in (9) is established in view of definition in (3) and using the elementary beta integral.
The Laplace transform of defined in (3), easily obtained here, is as follows:where ; and
Here, is the generalized Lauricella function ([21], p. 37, Equations (21–23)).
3. Main Results
Theorem 1. Let .
Then, for , we haveandIf with the initial condition ( is an arbitrary constant) and solution of differential equations existing in the space , then Theorems 2–4 are stated in the following form.
Theorem 2. Ifthen its solution is given by
Theorem 3. Ifthen its solution is given by
Theorem 4. Ifthen its solution is given by
Proof. In Theorem 1, let the left-hand side of result (11) be , i.e.,Having used the definition of given in (3), we obtain the following form:On using the fractional derivative of power function ([20], p. 36, Equation (2.26)), we haveOn interpreting multiple series by the definition of , we at once arrive at (11).
The proof of (12) follows the proof of (11) using (6) and ([20], p. 40, Equation (2.44)) therein.
Theorem 2 is proved as follows.
Use the definition of operator (at and ) and result (9) (at ) in (13), we haveTaking Laplace transform of (22) and then using formula (7) (for n = 1) and (10), therein we haveIn view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we have the formApplying inverse Laplace transform on both sides of (24) and using convolution theorem, we findNow, on interpreting the multiple series using (3), we at once arrive at desired result (14).
Proof. of Theorem 3. We use (at and ) and (9) (at ) in (15), and it takes the following form:On both sides of (26), we take Laplace transform and then using formula (7) (for n = 1) and (10) therein, we obtainIn view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we haveApplying inverse Laplace transform on (28), we haveOn interpreting the multiple series using (3), we at once arrive at result (16).
Proof. of Theorem 4. We use operator (at and ) and (9) (at ) in (17), and we have the following form:On both sides of (30), we take Laplace transform and use formulae (8) and (10) (for n = 1); then, we obtainIn view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we haveSince this is a linear differential equation of first order and first degree,Taking inverse Laplace transform of (33), we haveIn view of convolution theorem, we obtainNow, on interpreting the multiple series in view of (3), we obtain the result in (18).
4. Conclusion
Here, we conclude further interesting known results:(1)Our main results for , respectively, give the known results provided by Gupta and Jaimini ([22], pp. 145–146, Equations (1–10)).(2)If in result (10) and in Theorem 2, we take , then result (10) reduces to the known result provided by Saxena et al. ([23], p. 10, Equation (50)) and Theorem 2 gives the correct form (at ) of the theorem provided by Saxena et al. ([23], p. 10, Theorem (5.1)).(3)For and , Theorems 1 to 4 reduce, respectively, to the known results (at ) provided by Srivastava and Tomovski [24].(4)If in Theorem 1, we take and , then these results, respectively, reduce to the known results provided by Gautam ([18], pp. 201–202, Equations (4.64)–(4.65)).(5)If in Theorems 1 to 4, we take , , and , then these, respectively, reduce to the results for function provided by Gupta ([25], pp. 250–253, Equations (4.9.19)–(4.9.27)).
Therefore, the results presented in the article would immediately yield a large number of results that include a wide range of special functions occurring in issues of scientific research, computer science, and applied mathematics, among others.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this article.