Some Fractional Operators with the Generalized Bessel–Maitland Function

Abstract

In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator and its inverse operator , involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.

1. Introduction

During the last few years, many types of research studies developed the class of generalized fractional integrals containing a variety of special functions [1–5]. Watson [6] discussed applications of the Bessel function with some fields of applied sciences, biology, chemistry, physical sciences, and engineering. The generalization and extensions of the Bessel–Maitland function [7–14] dealt with special cases that gave useful results in different areas of mathematics. The recent work in the field of fractional calculus theory, differential equations of the Mittag-Leffler function, Sturm–Liouville problems in theoretical sense, Gronwall’s inequality, and exponential kernels of the differential operator [15–18] have found many applications in various subfields of mathematical analysis.

The series representation of the Bessel–Maitland function [19] is defined as

The generalization of the Bessel–Maitland function introduced by Singh et al. [7] iswhere , , and .

The extended Bessel–Maitland function investigated in [20] iswhere , , , and .

The Saigo fractional integral operators are defined [21] for , , and :

Samko et al. [22] defined the Riemann–Liouville fractional operators for and as

The Gauss hypergeometric function defined by Saigo [21] for all , , and iswhere are Pochhammer’s symbols.

Pochhammer’s symbols defined by Petojevic [23] arewhere and , and in gamma form, they can be written as

The beta function is defined as given in [23], for , and also expressed in the gamma form, respectively:

The gamma function is defined [23] for as

The generalized hypergeometric function is defined by Rainville [24]:where , .

The generalized Fox–Wright function is defined as [25]where , , and .

The Gauss hypergeometric function in the gamma form can be written as

The Laplace transform of function is defined as

Dirichlet formula (Fubini’s theorem) [22] is given by

Definition 1. The generalization of the generalized Bessel–Maitland function is defined and investigated aswhere , , , , , , , and .
The following notation is used in our results:

Definition 2. The extension of generalized Bessel–Maitland function (19) in multivariable function can be defined for , , , , , , , and , , as

Remark 1. On setting in equation (21), we get generalized Bessel–Maitland function (19).

Definition 3. An integral operator which involves generalized Bessel–Maitland function (19) as its kernel is defined for , , , , , , , , and as follows:

Remark 2. If we put and replace by , then it will become a left-sided Riemann–Liouville fractional integral operator.
The new fractional operator (22) can be discussed to improve the results of some inequalities such as Polya–Szego inequality, Chebyshev inequality, and Hadamard inequality in the field of analysis.

Definition 4. The left inverse operator of integral operator (22), for , , , , , , , , , and as is defined as follows:

Remark 3. If we put and replace by , then equation (23) becomes the Riemann–Liouville fractional differential operator.

Remark 4. If we replace , , , and in equation (23), we getwhere the inverse operator is described and discussed by Polito and Tomovski in [26].

2. Relation with the Bessel–Maitland and the Mittag-Leffler Functions

In this section, we discuss some special cases of the generalized Bessel–Maitland function and developed its relations with generalized Mittag-Leffler functions:(i)On replacing in equation (19), we obtain the relation where is the generalized Bessel–Maitland function investigated in [20].(ii)On replacing in equation (19), we obtain the relation where is the generalization of Bessel–Maitland function defined by Singh et al. [7].(iii)On replacing and in equation (19), we obtain the relation [19](iv)On setting and replacing by in equations (19) and (22), then we have where is the Mittag-Leffler function investigated by Salim and Faraj [27].(v)On setting and and replacing by in equation (19), then we have where is the Mittag-Leffler function defined by Shukla and Prajapati [28] and is described by Srivastava and Tomovski in [29].(vi)On setting and and replacing by in equations (19) and (22), then we have where is defined by Prabhakar in [30] and described the integral(vii)On setting and and replacing by in equation (19), then we have where is the Mittag-Leffler function discussed by Wiman [31].(viii)On setting , , and in equation (19), then we have where is the Mittag-Leffler function introduced in [32].

3. Convergence and Boundedness of the New Fractional Integral Operator

In this section, we discuss the convergence and boundedness of the fractional integral operator involving the generalized Bessel–Maitland function as its kernel in the form of a theorem.

Theorem 1. Let the operator be bounded on with , , , , , , , and ; then, the following relation holds:where

Proof. Let denote the th term of (36); then,Hence, as , and which means that the right-hand side of (36) is convergent and finite under the given condition. The condition of boundedness of the integral operator is discussed in the space of Lebesgue measure of a continuous function on , where :According to equations (19) and (22), we haveBy putting the values , , and in equation (39), we haveTherefore,

4. The Generalized Bessel–Maitland Function with Some Fractional Integral Operators

In this section, we derive some results of Saigo fractional integral operators with the generalized Bessel-Maitland function, and these results are established in terms of the Fox–Wright function. Also, we develop the composition of Riemann–Liouville operators with the generalized Bessel–Maitland function.

Theorem 2. Let with , , , , , , , , and ; then, the following relation holds:

Proof. Consider the left-sided Saigo fractional integral operator (4), in which using the power function with generalized Bessel–Maitland function (19), we getBy using equation (8) in equation (43), we haveBy putting the values , , and in equation (45), we obtainBy using equations (11) and (12) in equation (46), we getBy using equations (10) and (16) in equation (47), we haveBy using equations (10) and (20) in equation (51), we getHence, we attain the required result:

Theorem 3. Let with , , , , , , , , and ; then, the following relation holds:

Proof. Consider the right-sided Sagio fractional integral operator (5), in which using the power function with generalized Bessel–Maitland function (19), we getBy using equation (8) in equation (52), we haveBy putting the values , , and in equation (53), we obtainBy using equations (11) and (12) in equation (54), we haveNow, by using equations (10) and (16) in equation (56), we getBy using equations (10) and (20) in (57), we have the result:

Theorem 4. Let , , , , , , , , and ; then, the following relation holds:

Proof. Consider the left-sided Riemann–Liouville fractional integral operator (6), in which using the power function with generalized Bessel–Maitland function (19), we getBy putting the values , , and in equation (59), we obtainBy using equations (11) and (12) in equation (60), we haveNow, by using equation (20) in equation (61), then the required result is obtained:

5. Riemann–Liouville Fractional Operators and Laplace Transform of the New Operator

In this section, we discuss the Riemann–Liouville fractional integral and differential operators with the fractional integral operator. Also, we developed a result which deals with the Laplace transform of the new fractional integral operator.

Theorem 5. Let , , , , , , , , , , , and ; then, the following relation holds:

Proof. Consider the left-sided Riemann–Liouville integral operator (6) involving new fractional integral operator (