## Abstract and Applied Analysis

### On Generalized Fractional Integral Operators and the Generalized Gauss Hypergeometric Functions

#### Abstract

A remarkably large number of fractional integral formulas involving the number of special functions, have been investigated by many authors. Very recently, Agarwal (National Academy Science Letters) gave some integral transform and fractional integral formulas involving the ${F}_{p}^{(\alpha ,\beta )}(·)$. In this sequel, here, we aim to establish some image formulas by applying generalized operators of the fractional integration involving Appell’s function ${F}_{3}(·)$ due to Marichev-Saigo-Maeda. Some interesting special cases of our main results are also considered.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 630840, 5 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412606729

Digital Object Identifier
doi:10.1155/2014/630840

Mathematical Reviews number (MathSciNet)
MR3200794

Zentralblatt MATH identifier
07022776

#### Citation

Baleanu, Dumitru; Agarwal, Praveen. On Generalized Fractional Integral Operators and the Generalized Gauss Hypergeometric Functions. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 630840, 5 pages. doi:10.1155/2014/630840. https://projecteuclid.org/euclid.aaa/1412606729

#### References

• P. Agarwal, M. Chand, and S. D. Purohit, “A note on generating functions involving generalized Gauss hypergeometric functionsčommentComment on ref. [2?]: Please update the information of this reference, if possible.,” National Academy Science Letters. In press.
• M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, “Extended hypergeometric and confluent hypergeometric functions,” Applied Mathematics and Computation, vol. 159, no. 2, pp. 589–602, 2004.
• E. Özergin, Some properties of hypergeometric functions, [Ph.D. thesis], Eastern Mediterranean University, North Cyprus, 2011.
• E. Özergin, M. A. Özarslan, and A. Alt\in, “Extension of gamma, beta and hypergeometric functions,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4601–4610, 2011.
• H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science, London, UK, 2012.
• P. Agarwal, “Certain properties of the generalized Gauss hypergeometric functions,” Applied Mathematics & Information Sciences, vol. 8, no. 5, pp. 2315–2320, 2014.
• D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Complexity, Nonlinearity and Chaos, World Scientific, 2012.
• D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “On the solution set for a class of sequential fractional differential equations,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 38, article 385209, 2010.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier (North-Holland) Science Publishers, Amsterdam, The Netherlands, 2006.
• V. Kiryakova, “On two Saigo's fractional integral operators in the class of univalent functions,” Fractional Calculus & Applied Analysis, vol. 9, no. 2, pp. 159–176, 2006.
• V. Kiryakova, “A brief story about the operators of the generalized fractional calculus,” Fractional Calculus & Applied Analysis, vol. 11, no. 2, pp. 203–220, 2008.
• O. I. Marichev, “Volterra equation of Mellin convolution type with a Horn function in the kernel,” Izvestiya Akademii Nauk BSSR. Seriya Fiziko-Matematicheskikh Nauk, no. 1, pp. 128–129, 1974 (Russian).
• A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, vol. 31 of Research Notes in Mathematics, Pitman, London, UK, 1979.
• M. Saigo, “On generalized fractional calculus operators,” in Recent Advances in Applied Mathematics, pp. 441–450, Kuwait University, Kuwait, 1996.
• M. Saigo and N. Maeda, “More generalization of fractional calculus,” in Transform Methods & Special Functions, pp. 386–400, IMI-BAS, Sofia, Bulgaria, 1998.
• H. M. Srivastava and P. Agarwal, “Certain fractional integral operators and the generalized incomplete hypergeometric functions,” Applied Mathematics and Computation, vol. 8, no. 2, pp. 333–345, 2013.
• 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, Cambridge University Press, Gaithersburg, Maryland, 2010.
• A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function : Theory and Applications, Springer, London, UK, 2010. \endinput