Journal of Mathematics

Volume 2018, Article ID 5198621, 8 pages

https://doi.org/10.1155/2018/5198621

## Generalized Fractional Integral Formulas for the -Bessel Function

^{1}Department of Mathematics, Wollo University, P.O. Box 1145, Dessie, Ethiopia^{2}Department of Physics, Wollo University, P.O. Box 1145, Dessie, Ethiopia

Correspondence should be addressed to D. L. Suthar; moc.liamg@rahtusld

Received 13 May 2018; Accepted 1 August 2018; Published 5 September 2018

Academic Editor: Sunil D. Purohit

Copyright © 2018 D. L. Suthar and Mengesha Ayene. 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.

#### 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.

#### References

- S. R. Mondal, Representation Formulae and Monotonicity of the Generalized k-Bessel Functions, 2016.
- Diaz and E. Pariguan, “On hypergeometric functions and Pochhammer k-symbol,”
*Divulgaciones Matemáticas*, vol. 15, no. 2, pp. 179–192, 2007. View at Google Scholar · View at MathSciNet - H. Satoh,
*Higher Transcendental Functions*, vol. 2, McGraw-Hill, NewYork, Toronto, London, 1953. View at Publisher · View at Google Scholar - P. Agarwal, M. Chand, J. Choi, and G. Singh, “Certain fractional integrals and image formulas of generalized k-Bessel function,”
*Communications of the Korean Mathematical Society*, vol. 33, no. 2, pp. 423–436, 2018. View at Google Scholar · View at MathSciNet - M. Chand, P. Agarwal, and Z. Hammouch, “Certain Sequences Involving Product of k-Bessel Function,”
*International Journal of Applied and Computational Mathematics*, vol. 4, no. 4, 4:101 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - K. S. Gehlot, “Differential Equation of K -Bessels Functions and its Properties,”
*Nonl. Analysis and Differential Equations*, vol. 2, no. 2, pp. 61–67, 2014. View at Publisher · View at Google Scholar - K. S. Gehlot, “Recurrence Relations of K -Bessels function,”
*Thai J. Math*. - K. S. Gehlot and S. D. Purohit, “Integral representations of the k-Bessel's function,”
*Honam Mathematical Journal*, vol. 38, no. 1, pp. 17–23, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - G. Singh, P. Agarwal, M. Chand, and S. Jain, “Certain fractional kinetic equations involving generalized,”
*Transactions of A. Razmadze Mathematical Institute*, 2018. View at Publisher · View at Google Scholar - M. Saigo and N. Maeda, “More generalization of fractional calculus,” in
*Transform Methods and Special Functions*, pp. 386–400, Bulgarian Acad. Sci., Sofia, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - O. I. Marichev, “Volterra equation of Mellin convolution type with a Horn function in the kernel,”
*Izvestiya Akademii Nauk BSSR, Seriya Fiziko-Matematicheskikh Nauk*, vol. 1, pp. 128-129, 1974 (Russian). View at Google Scholar - H. M. Srivastava and P. W. Karlsson,
*Multiple Gaussian hypergeometric series*, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, NY, USA, 1985. View at MathSciNet - F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, Eds.,
*NIST Handbook of Mathematical Functions, National Institute of Standards and Technology*, USA; Cambridge University Press, Gaithersburg, Md, 2010. - A. Erde'lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi,
*Higher Transcendental Functions*, vol. 1, McGraw-Hill, NewYork, Toronto, London, 1953. - E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,”
*Journal of the London Mathematical Society*, vol. 1-10, no. 4, pp. 286–293, 1935. View at Publisher · View at Google Scholar · View at Scopus - E. M. Wright, “The asymptotic expansion of the generalized hypergeometric function,”
*Proceedings of the London Mathematical Society*, vol. 46, no. 2, pp. 389–408, 1940. View at Google Scholar · View at MathSciNet - E. M. Wright, “The asymptotic expansion of integral functions defined by Taylor series,”
*Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences*, vol. 238, pp. 423–451, 1940. View at Publisher · View at Google Scholar · View at MathSciNet - A. A. Kilbas, M. Saigo, and J. J. Trujillo, “On the generalized Wright function,”
*Fractional Calculus and Applied Analysis*, vol. 5, no. 4, pp. 437–460, 2002. View at Google Scholar · View at MathSciNet - M. Saigo, “A remark on integral operators involving the Gaüss hypergeometric functions,”
*Mathematical Reports of College of General Education, Kyushu University*, vol. 11, no. 2, pp. 135–143, 1978. View at Google Scholar · View at MathSciNet - A. A. Kilbas and N. Sebastian, “Generalized fractional integration of Bessel function of the first kind,”
*Integral Transforms and Special Functions*, vol. 19, no. 11-12, pp. 869–883, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Amsalu and D. L. Suthar, “Generalized Fractional Integral Operators Involving Mittag-Leffler Function,”
*Abstract and Applied Analysis*, vol. 2018, Article ID 7034124, 8 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - S. R. Mondal and K. S. Nisar, “Marichev-Saigo-Maeda Fractional Integration Operators Involving Generalized Bessel Functions,”
*Mathematical Problems in Engineering*, vol. 2014, Article ID 274093, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - K. S. Nisar, D. L. Suthar, S. D. Purohit, and M. Aldhaifallah, “Some unified integrals associated with the generalized Struve function,”
*Proceedings of the Jangjeon Mathematical Society. Memoirs of the Jangjeon Mathematical Society*, vol. 20, no. 2, pp. 261–267, 2017. View at Google Scholar · View at MathSciNet - S. D. Purohit, D. L. Suthar, and S. L. Kalla, “Marichev-Saigo-Maeda fractional integration operators of the Bessel functions,”
*Le Matematiche*, vol. 67, no. 1, pp. 21–32, 2012. View at Google Scholar · View at MathSciNet - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives, Theory and Applications*, Gordon and Breach, Yverdon, Switzerland, 1993. View at MathSciNet - D. L. Suthar and H. Habenom, “Integrals involving generalized Bessel-Maitland function,”
*Journal of Science and Arts*, no. 4(37), pp. 357–362, 2016. View at Google Scholar · View at MathSciNet