The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 12.3436 11.6718" width="12.3436pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-87" d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"/></g></svg>-</span>Function

Chandak, S.; Shimelis, Biniyam; Abeye, Nigussie; Padma, A.

doi:https://doi.org/10.1155/2021/9961013

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2021 | Article ID 9961013 | https://doi.org/10.1155/2021/9961013

The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the -Function

S. Chandak,¹Biniyam Shimelis,²Nigussie Abeye,²and A. Padma³

Academic Editor: Victor Kovtunenko

Received25 Mar 2021

Revised13 Jun 2021

Accepted23 Jul 2021

Published12 Aug 2021

Abstract

In the present paper, we establish some composition formulas for Marichev-Saigo-Maeda (MSM) fractional calculus operators with -function as the kernel. In addition, on account of -function, a variety of known results associated with special functions such as the Mittag-Leffler function, exponential function, Struve’s function, Lommel’s function, the Bessel function, Wright’s generalized Bessel function, and the generalized hypergeometric function have been discovered by defining suitable values for the parameters.

1. Introduction and Preliminaries

Fractional calculus is very old and similar to conventional calculus that has gradually been applied to a variety of fields including technology, science, economics, arithmetic geometry, and computer science. The -functions are especially valuable functions that provide solutions to a variety of problems developed in terms of fractional order differential, integral, and difference equations. As a result of that, this field has become an area of interest for researchers in recent times. In addition, a majority of scholars (see [1–3]) have investigated the properties, uses, and various extensions of a variety of fractional calculus operators in depth. Also, on other similar themes, there is a lot of activity and coverage all over the world. The research monographs [4, 5] can be consulted.

Kumar [6] has recently characterized the -function as follows: where (1) are real numbers(2), , and are natural numbers,(3), is a complex variable and is an arbitrary constant(4)The series on the RHS of (1) converges absolutely if or with

For descriptions of the series convergence constraints on (1) RHS, simply review [7–10]. The -function defined by (1) is of a general character since it assimilates and applies a variety of valuable functions such as Macrobert’s -function and the exponential function [11], the generalized Mittag-Leffler function [12–15], Lommel’s function [16], Struve’s function [17–21], the Wright generalized Bessel function and the Bessel function [22–25], the generalized hypergeometric function [11, 26], and the unified Riemann-Zeta function [27].

Special cases of (1) are as follows: (i)If we put , and in equation (1) then the -function turns into the generalized hypergeometric function [11]:(ii)If we put and in equation (1) then the -function turns into the Bessel function [23]:(iii)If we put , and in equation (1) then the -function turns into Wright’s generalized Bessel function [23]:(iv)If we put , and in equation (1) then the -function turns into Struve's function [23]:(v)If we put , and in equation (1) then the -function turns into Lommel’s function [23]:(vi)If we put , and in equation (1) then the -function turns into Mittag-Leffler function [12, 13]:(vii)If we put , and in equation (1) then the -function turns into unified Riemann-Zeta function [27]:(viii)If we put , and in equation (1) then the -function turns into the function:(ix)If we put , and in equation (1) then the -function turns into Macrobert's -function [11]:(x)If we put and in equation (1) then the -function turns into the function:(xi)If we put , and in equation (1) then the -function turns into the function:

1.1. Saigo Fractional Calculus Operators

The fractional integral and differential operators with the Gauss hypergeometric function as the kernel are characterized by Saigo [28], and they are notable generalizations of the Riemann-Liouville (R-L) and Erdélyi-Kober fractional calculus operators (see [1]).

For and with , the left-hand and the right-hand sided generalized fractional integral operators connected with Gauss hypergeometric function are defined as below: respectively. Here, is the Gauss hypergeometric function [1] defined for and by where The corresponding fractional differential operators are where and is the integer part of . Substituting and in equations (14)–(17), we get the corresponding R-L and Erdélyi-Kober fractional operators, respectively.

1.2. MSM Fractional Calculus Operators

Marichev [29] introduced and researched fractional calculus operators, which are an extension of the Saigo operators, which were later expanded by Saigo and Maeda [30]. For and with , the left-hand and right-hand sided MSM fractional integral and derivative operators associated with third Appell function are defined as respectively, where , and the third Appell function [29] is defined by

The following MSM integral operators are required here [[30], p. 394] to obtain the MSM fractional integration of the generalized -function.

Lemma 1. Let such that (i) then(i)If then

Fractional integral formulas for the -function are given by [10]. In order to provide unification and extension of such kinds of results on fractional integrals of special functions. The authors derive two unified fractional integrals involving the product of -function [6] and Appell function [29]. The integrals are further used in establishing two theorems on Saigo-Maeda operators of fractional integration [30].

In this paper, our aim is to study the compositions of the generalized fractional integration operators (18) and (19) with the -function (1). At least, we establish them as special cases on the main result in connection with various special functions.

2. Fractional Calculus Approach

In this section, we establish image formulas for the -function involving the left and right sides of the MSM fractional integral operators. These formulas are given by the following theorems.

Theorem 2. Let ; ; ; , and is an arbitrary constant, such that , then for following integral holds true:

Proof. For convenience, we denote the left-hand side of the result (25) by using the definitions (1) and taking the left-hand sided MSM fractional integral operator inside the summation, the left-hand side of (25) becomes by applying the relation (23) in (26), we get RHS of (25).☐

The next theorem gives the right-hand MSM fractional integration of the -function.

Theorem 3. Let; ; ; , , andis an arbitrary constant, such that, then forfollowing integral holds true:

Proof. For convenience, we denote the left-hand side of the result (27) by using the definitions (1) and taking the right-hand sided MSM fractional integral operator inside the summation, the left-hand side of (27) becomes by applying relation (24) in (28), we get R.H.S. of (27).☐

3. Special Cases

Here, we present some special cases by choosing suitable values of the parameters in Theorems 3.1 and 2.2. We get certain interesting results concerning the other special functions like the generalized hypergeometric function, the Bessel function, Wright’s generalized Bessel function, Struve’s function, Lommel’s function, the Mittag-Leffler function, the Riemann-Zeta function, the exponential function, and Macrobert’s E-function given in the following special cases. (i)If we put , and in equations (25) and (27), then the -function turns into the generalized hypergeometric function [11] and applying the result from ([31], equations (25) and (26), page 34)).

As per condition of Theorems 3.1 and 2.2 respectively as below: (ii)If we put , and in equations (25) and (27) then the -function turns into the Bessel function [23] and applying the result from ([31], equations (25) and (5.3), page 38) asunder the condition stated in Theorems 3.1 and 2.2, respectively, as below: (iii)If we put and in equations (25) and (27) then the -function turns into Wright’s generalized Bessel function [23] using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get(iv)If we put , and in equation (25) and (27) then V-function turn into Struve's function [23] using the result (29) under the condition in Theorem 3.1 and 2.2 respectively, we get(v)If we put and in equations (25) and (27) then V-function turn into Lommel’s function [23] using result (29) under the condition in Theorems 3.1 and 2.2, respectively. we get(vi)If we put and in equation (25) and (27) then the -function turns into the Mittag-Leffler function [12, 13] using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get(vii)If we put , and in equations (25) and (27) then the -function turns into the unified Riemann-Zeta function [27] using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get,(viii)If we put , and in equations (25) and (27) then V-function turn into function, using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get,(ix)If we put and in equations (25) and (27) then the -function turns into Macrobert’s -function [11] using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get(x)If we put , and in equations (25) and (27) then the -function turns into function, using result (29) under the condition in Theorems 3.1 and 2.2, respectively, we get(xi)If we put and in equation (25) and (27) then the -function turns into function, using result (29) under the condition in Theorems 3.1 and 2.2, respectively. We get

4. Concluding Remark

Due to the generalization of Riemann-Liouville, Weyl, Erdélyi-Kober, and Saigo’s fractional calculus operators, MSM fractional calculus operators have a significant advantage; as a result, many writers are referred to as general operators. Now, we will wrap up this paper by highlighting that our most important findings (Theorems 2 and 3) can be deduced as special cases involving well-known fractional calculus operators, as previously stated. The -function defined in (1), on the other hand, has the property that a number of special functions appear to be the special cases. Several special cases involving integrals relating to the -function have been uncovered in previous research works by various authors using different arguments.

Data Availability

No data were used to support this study

Conflicts of Interest

There is no conflict of interest regarding the publication of this article.

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Copyright © 2021 S. Chandak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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