## Abstract and Applied Analysis

### Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

#### Abstract

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving the ${F}_{p}^{(\alpha ,\beta )}$ $(·)$. In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functions ${F}_{p}^{(\alpha ,\beta ,m)}$ $(·)$. Some interesting special cases of our main results are also considered.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 735946, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278549

Digital Object Identifier
doi:10.1155/2014/735946

Mathematical Reviews number (MathSciNet)
MR3226228

Zentralblatt MATH identifier
07022978

#### Citation

Choi, Junesang; Agarwal, Praveen. Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions. Abstr. Appl. Anal. 2014 (2014), Article ID 735946, 7 pages. doi:10.1155/2014/735946. https://projecteuclid.org/euclid.aaa/1412278549

#### References

• H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science, Amsterdam, The Netherlands, 2012.
• P. Agarwal, M. Chand, and S. D. Purohit, “A note on generating functions involving generalized GaussčommentComment on ref. [7?]: Please update the information of this reference, if possible. hypergeometric functions,” National Academy Science Letters. In press.
• M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, “Extension of Euler\textquotesingle s beta function,” Journal of Computational and Applied Mathematics, vol. 78, no. 1, pp. 19–32, 1997.
• 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.
• D. M. Lee, A. K. Rathie, R. K. Parmar, and Y. S. Kim, “Generalization of extended beta function, hypergeometric and confluent hypergeometric functions,” Honam Mathematical Journal, vol. 33, no. 2, pp. 187–206, 2011.
• E. Özergin, Some properties of hypergeometric functions [Ph.D. thesis], Eastern Mediterranean University, Gazimağusa, 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. Liu and W. Wang, “Some generating relations for extended Appell\textquotesingle s and Lauricella\textquotesingle s hypergeometric functionsčommentComment on ref. [16?]: Please update the information of this reference, if possible.,” The Rocky Mountain Journal of Mathematics. In press.
• R. K. Parmar, “A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions,” Le Matematiche, vol. 68, no. 2, pp. 33–42, 2013.
• H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press, Ellis Horwood Limited, Chichester, UK, John Wiley and Sons, New York, NY, USA, 1985.
• P. Agarwal, “Certain properties of the generalized Gauss hypergeometric functions,” Applied Mathematics & Information Sciences, vol. 8, no. 5, pp. 2315–2320, 2014.
• I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill, New Delhi, India, 1979.
• P. Agarwal, “Fractional integration of the product of two multivariables $H$-function and a general class of polynomials,” in Advances in Applied Mathematics and Approximation Theory, vol. 41 of Springer Proceedings in Mathematics & Statistics, pp. 359–374, Springer, New York, NY, USA, 2013.
• P. Agarwal, “Further results on fractional calculus of Saigo operators,” Applications and Applied Mathematics, vol. 7, no. 2, pp. 585–594, 2012.
• P. Agarwal, “Generalized fractional integration of the $\overline{H}$ function,” Le Matematiche, vol. 67, no. 2, pp. 107–118, 2012.
• P. Agarwal and S. Jain, “Further results on fractional calculus of Srivastava polynomials,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 167–174, 2011.
• P. Agarwal and S. D. Purohit, “The unified pathway fractional integral formulae,” Journal of Fractional Calculus and Applications, vol. 4, no. 9, pp. 1–8, 2013.
• A. A. Kilbas, “Fractional calculus of the generalized Wright function,” Fractional Calculus & Applied Analysis, vol. 8, no. 2, pp. 113–126, 2005.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematical Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
• V. Kiryakova, “On two Saigo\textquotesingle s fractional integral operators in the class of univalent functions,” Fractional Calculus & Applied Analysis, vol. 9, no. 2, pp. 160–176, 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.
• S. Kumar, D. Kumar, S. Abbasbandy, and M. M. Rashidi, “Analytical solution of fractional Navier-Stokes equation byusing modifed Laplace decomposition method,” Ain Shams Engineering Journal, vol. 5, pp. 569–574, 2014.
• M. Saigo, “On generalized fractional calculus operators,” in Proceedings of the International Workshop on Recent Advances in Applied Mathematics, pp. 441–4450, Kuwait University, Kuwait, 1996.
• M. Saigo, “A remark on integral operators involving the Gauss hypergeometric functions,” Mathematical Reports of College of General Education. Kyushu University, vol. 11, no. 2, pp. 135–143, 1978.
• M. Saigo, “A certain boundary value problem for the Euler-Darboux equation I,” Mathematica Japonica, vol. 24, no. 4, pp. 377–385, 1979.
• M. Saigo and N. Maeda, “More generalization of fractional calculus,” in Proceedings of the 2nd International Workshop on Transform Metods and Special Functions, P. Rusev, I. Dimovski, and V. Kiryakova, Eds., pp. 386–400, IMI, Varna, Bulgaria, August 1996.
• H. M. Srivastava and P. Agarwal, “Certain fractional integral operators and the generalized incomplete hypergeometric functions,” Applications and Applied Mathematics, vol. 8, no. 2, pp. 333–345, 2013.
• A.-M. Yang, Y.-Z. Zhang, C. Cattani et al., “Application of local fractional series expansion method to solve Klein-Gordon equations on Cantor sets,” Abstract and Applied Analysis, vol. 2014, Article ID 372741, 6 pages, 2014. \endinput