## Fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function.(English)Zbl 1474.26030

Summary: We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.

### MSC:

 26A33 Fractional derivatives and integrals 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)
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### References:

 [1] Agarwal, P., Further results on fractional calculus of Saigo operators, Applications and Applied Mathematics: An International Journal (AAM), 7, 2, 585-594, (2012) · Zbl 1256.26003 [2] Agarwal, P.; Jain, S., Further results on fractional calculus of Srivastava polynomials, Bulletin of Mathematical Analysis and Applications, 3, 2, 167-174, (2011) · Zbl 1314.26005 [3] Kalla, S. L.; Saxena, R. K., Integral operators involving hypergeometric functions, Mathematische Zeitschrift, 108, 231-234, (1969) · Zbl 0169.14503 [4] Kilbas, A. A., Fractional calculus of the generalized Wright function, Fractional Calculus & Applied Analysis, 8, 2, 113-126, (2005) · Zbl 1144.26008 [5] Kilbas, A. A.; Sebastian, N., Generalized fractional integration of Bessel function of the first kind, Integral Transforms and Special Functions, 19, 11-12, 869-883, (2008) · Zbl 1156.26004 [6] Kiryakova, V., A brief story about the operators of the generalized fractional calculus, Fractional Calculus & Applied Analysis, 11, 2, 203-220, (2008) · Zbl 1153.26003 [7] Love, E. R., Some integral equations involving hypergeometric functions, Proceedings of the Edinburgh Mathematical Society, 15, 3, 169-198, (1967) · Zbl 0173.14202 [8] Nisar, K. S.; Suthar, D. L.; Purohit, S. D.; Aldhaifallah, M., Some unified integrals associated with the generalized Struve function, Proceedings of the Jangjeon Mathematical Society. Memoirs of the Jangjeon Mathematical Society, 20, 2, 261-267, (2017) · Zbl 1371.33013 [9] Purohit, S. D.; Suthar, D. L.; Kalla, S. L., Marichev-Saigo-Maeda fractional integration operators of the Bessel functions, Le Matematiche, 67, 1, 21-32, (2012) · Zbl 1247.26012 [10] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations, (2006), Amsterdam, Netherlands: Elsevier, Amsterdam, Netherlands · Zbl 1092.45003 [11] Mathai, A. M.; Saxena, R. K.; Haubold, H. J., The H-function : Theory and Applications, (2010), New York, NY, USA: Springer, New York, NY, USA · Zbl 1181.33001 [12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, (1993), Yverdon, Switzerland: Gordon and Breach, Yverdon, Switzerland · Zbl 0818.26003 [13] Srivastava, H. M.; Saxena, R. K., Operators of fractional integration and their applications, Applied Mathematics and Computation, 118, 1, 1-52, (2001) · Zbl 1022.26012 [14] Kiryakova, V. S., Generalized fractional calculus and applications, Research Notes in Mathematics Series, (1993), New York, NY, USA: Longman, Harlow; Wiley, New York, NY, USA [15] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002 [16] Saigo, M., A remark on integral operators involving the Gaüss hypergeometric functions, Mathematical Reports of College of General Education, Kyushu University, 11, 2, 135-143, (1978) [17] Saigo, M., A certain boundary value problem for the Euler-Darboux equation, Mathematica Japonica, 24, 4, 377-385, (1979) · Zbl 0428.35064 [18] Saigo, M.; Maeda, N., More generalization of fractional calculus, Transform Methods and Special Functions, 386-400, (1998), Bulgarian Acad. Sci., Sofia · Zbl 0926.26003 [19] Marichev, O. I., Volterra equation of Mellin convolution type with a Horn function in the kernel, Izvestiya Akademii Nauk BSSR, Seriya Fiziko-Matematicheskikh Nauk, 1, 128-129, (1974) [20] Srivastava, H. M.; Karlsson, P. W., Multiple Gausian Hypergeometric Series, (1985), New York, NY, USA: Halsted Press, Chichester; Wiley, New York, NY, USA [21] Kober, H., On fractional integrals and derivatives, Quarterly Journal of Mathematics, 11, 1, 193-211, (1940) · JFM 66.0520.02 [22] Oldham, K. B.; Spanier, J., The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, (1974), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0292.26011 [23] Saxena, R. K.; Saigo, M., Generalized fractional calculus of the H-function associated with the Appell function F3, Journal of Fractional Calculus and Applications, 19, 89-104, (2001) · Zbl 0984.33006 [24] Saxena, R. K.; Daiya, J., Integral transforms of the S-functions, Le Matematiche, 70, 2, 147-159, (2015) · Zbl 1342.26024 [25] Diaz, R.; Pariguan, E., On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Matemáticas, 15, 2, 179-192, (2007) · Zbl 1163.33300 [26] Romero, L. G.; Cerutti, R. A., Fractional Fourier transform and special k-functions, International Journal of Contemporary Mathematical Sciences, 7, 13-16, 693-704, (2012) · Zbl 1248.42006 [27] Saxena, R. K.; Daiya, J.; Singh, A., Integral transforms of the k-generalized Mittag-Leffler function, Le Matematiche, 69, 2, 7-16, (2014) · Zbl 1318.33037 [28] Sharma, K., Application of fractional calculus operators to related areas, General Mathematics Notes, 7, 1, 33-40, (2011) [29] Sharma, M.; Jain, R., A note on a generalized M-series as a special function of fractional calculus, Fractional Calculus and Applied Analysis, 12, 4, 449-452, (2009) · Zbl 1196.26013 [30] Wright, E. M., The asymptotic expansion of the generalized hypergeometric function, Journal of the London Mathematical Society, 1-10, 4, 286-293, (1935) · JFM 61.0407.01 [31] Wright, E. M., The asymptotic expansion of integral functions defined by Taylor series, Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, 238, 423-451, (1940) · JFM 66.0352.01 [32] Erde’lyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of Integral Transforms, 1, (1953), New York - Toronto - London: McGraw-Hill, New York - Toronto - London [33] Sneddon, I. N., The Use of Integral Transforms, (1979), New Delhi, India: Tata McGraw-Hill, New Delhi, India · Zbl 0433.73015 [34] Schiff, J. L., The laplace Transform: Theory and Applications, (1999), New York, NY, USA: Springer, New York, NY, USA · Zbl 0934.44001 [35] Amsalu, Hafte; Suthar, D. L., Generalized fractional integral operators involving mittag-leffler function, Abstract and Applied Analysis, 2018, (2018) · Zbl 1470.44002 [36] Mishra, V. N.; Suthar, D. L.; Purohit, S. D.; Srivastava, H. M., Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function, Cogent Mathematics, 4, 1-11, (2017) [37] Purohit, S. D.; Kalla, S. L.; Suthar, D. L., Fractional integral operators and the multiindex Mittag-Leffler functions, Scientia. Series A. Mathematical Sciences. New Series, 21, 87-96, (2011) · Zbl 1250.26007 [38] Purohit, S. D.; Suthar, D. L.; Kalla, S. L., Some results on fractional calculus operators associated with the M-function, Hadronic Journal, 33, 3, 225-235, (2010) · Zbl 1218.26003 [39] Saxena, R. K.; Ram, J.; Suthar, D. L., Fractional calculus of generalized Mittag-Leffler functions, Indian Academy of Mathematics, 31, 1, 165-172, (2009) · Zbl 1218.26005 [40] Suthar, D. L.; Habenom, H., Certain generalized fractional integral formulas involving the product of K-function and the general class of multivariable polynomials, Communications in Numerical Analysis, 2, 101-108, (2017) [41] Suthar, D. L.; Habenom, H.; Tadesse, H., Generalized fractional calculus formulas for a product of Mittag-Leffler function and multivariable polynomials, International Journal of Applied and Computational Mathematics, 4, 1, (2018) · Zbl 1380.26008
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