Abstract
The aim of this paper is to deal with two integral transforms involving the Appell function as their kernels. We prove some compositions formulas for generalized fractional integrals with -Bessel function. The results are expressed in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some related assertion for Saigo, Riemann-Liouville type, and Erdélyi-Kober type fractional integral transforms.
1. Introduction and Preliminaries
The generalization of -Bessel function is defined in Mondal [1] aswhere , , and and is the -gamma function defined in Dáaz and Pariguan [2] asBy inspection the following relation holds:andIf and , then the generalized -Bessel function defined in (1) reduces to the classical Bessel function defined in Erdélyi [3]. For more details regarding -Bessel function and its properties see [4–9].
Here, we establish various generalized fractional integral formulas for the -Bessel function. For this, we recall here the definition of generalized fractional integration operators of arbitrary order involving the Appell function ([10], p.393, eq. (4.12) and (4.13)) as in the kernel. A generalization of the hypergeometric fractional integrals for and is established by Marichev [11] as follows:In (5) and (6), denotes the Appell function (also well-known as Horn function) which is established by Srivastava and Karlsson [12].The properties of this function are studied in Olver et al. ([13], P. (412-415)). Further relation for the Gauss hypergeometric functions exists as follows:The generalized hypergeometric type function is represent in Erdélyi [14] aswhere ; , , , and is the Pochhammer symbols. Now, the gamma function is defined asand beta function is termed asThe following series is defined in the Wright type hypergeometric function (see [15–17]) aswhere and are positive real numbers, such that Equation (13) differs from the generalized hypergeometric function defined in (9) only by a constant multiplier. The generalized hypergeometric function is a special case of for , where and :For various properties of this function see [18].
2. Representation of Generalized Fractional Integrals in terms of Wright Functions
Marichev-Saigo-Maeda integrals operators were generalization of Saigo fractional integral operators [19]. In addition, their properties have been studied by Saigo and Maeda [10]. Considering this, the left-hand side and right-hand side of types (5) and (6) for a power function are as follows.
Lemma 1. (a) If , then(b) If , thenwhere .
The left-hand side generalized fractional integration (5) of the -Bessel functions (1) is given by the following result.
Theorem 2. Letting be such that , , then the following formula holds:
Proof. Using (1) and writing the function in the series form, the left-hand side of (18), leads to Now, upon using the image formula (16), which is valid under the condition declared with Theorem 2, we get Using the definition of (15) in the right-hand side of (20), we arrive at the result (18).
Special Cases of Theorem 2
(i) If we set and replace by in (18), then we get the following corollary relating to left-hand sided Saigo fractional integral operator ([19, 20]).
Corollary 3. Let , , and ; then the following formula holds:
(ii) If we set in (21), then we get the subsequent corollary relating to left-sided Riemann-Liouville type integral operator.
Corollary 4. Let be such that , , ; then the following result holds:
(iii) If we set in (21), then we get the subsequent corollary relating to left-sided Erdélyi-Kober type integral operator.
Corollary 5. Let be such that , ; then the following formula holds:
Theorem 6. Letting be such that , , then the following formula holds:
Proof. Using (2) and writing the function in the series form, the left-hand side of (24), leads to Now, upon using the image formula (17), which is valid under the conditions declared with Theorem 6, we get Using the definition of (15) in the right-hand side of (26), we arrive at the result (24).
Special Cases of Theorem 6
(iv) If we substitute and replace by in (24), then we get the subsequent corollary relating to right-hand sided Saigo fractional integral operator [19].
Corollary 7. Letting and , , , then the following formula holds:
(v) If we set in (27), then we get the following corollary relating to right-sided Weyl fractional type integral operator.
Corollary 8. Let be such that , , ; then the following result holds:
(vi) If we set in (27), then we get the subsequent corollary relating to right-hand side of Erdélyi-Kober fractional type integral operator.
Corollary 9. Let be such that , , ; then the following formula holds:
3. Representation in Terms of Generalized Hypergeometric Function
In this part, we introduce the generalized fractional integrals of -Bessel function in terms of generalized hypergeometric function. We consider the following well-known results:andWe represent the following theorems containing the generalized hypergeometric function.
Theorem 10. Let be such that , , , and let ; then the following formula holds:
Proof. Note that defined in (32) exit as the series is absolutely convergent. Now, using (11) with and (20) and applying (31) with being replaced by , and , we have Thus, in accordance with (9), we get the required result (32).
Corollary 11. Let be such that , , and let ; then the following result holds:
Corollary 12. Let be such that , , and ; then the following result holds:
Corollary 13. Let be such that , , and let ; then the following result holds:
Theorem 14. Letting be such that , , , then there holds the following formula:
Proof. Using (11) with and (26) and applying (31) with being replaced by , , and , we have Thus, in accordance with (9), we get the required result (37).
Corollary 15. Let and be such that , , and let ; then the following result holds:
Corollary 16. Let and be such that , , and let ; then the following result holds:
Corollary 17. Let and be such that , , and let ; then the following formula holds:
4. Concluding Remark
MSM fractional integral operators have advantage that they generalize the R-L, Weyl, Erdélyi-Kober, and Saigo’s fractional integral operators; therefore, several authors called this a general operator. So, we conclude this paper by emphasizing that many other interesting image formulas can be derived as the specific cases of our leading results, Theorems 2 and 6, involving familiar fractional integral operators as above said. Further, the generalized Bessel function defined in (1) possesses the lead that a number of Bessel functions, trigonometric functions, and hyperbolic functions happen to be the particular cases of this function. Some special cases of integrals involving generalized Bessel function have been explored in the literature by a number of authors ([20–26]) with different arguments. Therefore, results presented in this paper are easily converted in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.
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.