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Generalized Fractional Calculus Operators Associated with K-function

D.L. Suthar

Abstract

The aim of this paper is to study some properties of K-function introduced by Sharma. Here we establish two theorems which give the image of this K-function under the generalized fractional integral operators involving Fox’s H-function as kernel. Corresponding assertions in term of Euler, Whittaker and K-transforms are also presented. On account of general nature of H-function and K-function a number of results involving special functions can be obtained merely by giving particular values for the parameters.


Keywords

Generalized fractional integral; K-function; H-function; Euler trans-form; Whittaker transforms; K-transform; Beta function

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References

P. Agarwal, Q. Al-Mdallal, Y.J. Cho, and S. Jain, "Fractional differential equations for the generalized Mittag-Leffler function", Advances in Difference Equations, Vol. 2018, no.1, 2018, Art. ID 58.

P. Agarwal, J.J. Nieto, M.J. Luo, "Extended Riemann-Liouville type fractional derivative operator with applications", Open Mathematics, Vol. 15, no 1, 1667-1681, 2017.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, "Higher Transcendental Functions", Vol.2, McGraw-Hill, New York, 1954.

C. Fox, "The G and H functions as symmetrical Fourier Kernels", Trans. Amer. Math. Soc., vol. 98, no. 3, pp.395-429, 1961.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, "Theory and Applications of Fractional Differential Equations", Elsevier, North Holland Math. Studies 204. Amsterdam, 2006.

I.O. Kymaz, A. Cetinkaya and P. Agarwal, "An extension of Caputo fractional derivative operator and its applications", Journal of Nonlinear Science and Applications, Vol. 9, no. 6, 3611-3621, 2016.

A.M. Khan, P. Ramani, D.L. Suthar and D. Kumar , "A note on 4 k fractional integral operator, Int. J. Appl. Comput. Math., Vol. 4, no. 1, 1-12, 2018.

G.M. Mittag-Leffler, "Sur la nuovelle function E (x)  ", C. R. Acad. Sci. Paris,, vol. 137, no. 2, pp. 554-558, 1903.

G.M. Mittag-Leffler, "Sur la representation analytique de’une branche uniforme une function monogene, Acta.Math., vol. 29, pp. 101-181, 1905.

A.M. Mathai and R.K. Saxena, "The H-functions with Applications in Statistics and other Disciplines", John Wiley and Sons, 1974.

A.M. Mathai, R.K. Saxena and H.J. Haubold, "The H-function Theory and Application", Springer, New York, 1954.

T.R. Prabhakar, "A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel", Yokohama Math. J., vol. 19, pp. 7-15, 1971.

S.D. Purohit, D.L. Suthar and S.L. Kalla, "Some results on fractional calculus operators associated with the M-function", Hadronic J., vol. 33, no. 3, pp. 225-236, 2010.

S.D. Purohit, D.L. Suthar and S.L. Kalla, "Marichev-Saigo-Maeda Fractional Integration Operators of the Bessel Functions", Matematiche (Catania), vol. 67, no. 1, pp. 12-32, 2012.

E.D. Rainville, "Special Functions", Chelsea Publishing Company, Bronx, New York, 1960.

R.K. Saxena and R.K. Kumbhat, "Integral operators involving H-function", Indian J. Pure Appl. Math., vol. 5, pp. 1-6, 1974.

R.K.Saxena, J. Ram and D.L. Suthar, "Generalized fractional calculus of the generalized Mittag-Leffler functions",J. Indian Acad. Math., vol.31, no. 1, pp. 165-172, 2009.

K. Sharma, "Application of fractional calculus operators to related area", Gen. Math. Notes, vol. 7, no.1, pp. 33-40, 2011.

I.N. Sneddon, "The use of Integral Transform", New Delhi, Tata McGraw Hill, 1979.

H.M. Srivatava, K.C. Gupta and S.P. Goyal, "The H-Function of One and Two Variable With Applications", South Asian Publications Pvt. Ltd, New Delhi, Madras, 1982.

D.L. Suthar and Haile Habenom Anteneh, "Certain generalized fractional integral formulas involving the product of K-function and the general class of multivariable polynomials", Commun. Numer. Anal., vol. 2017, no. 2, pp.101-108, 2017.

D.L. Suthar and H. Habenom and H. Tadesse, "Generalized fractional calculus formulas for a product of Mittag-Leffler function and multivariable polynomials," Int. J. Appl. Comput. Math., Vol. 4, no. 1, 1-12, 2018.

D.L. Suthar and S.D. Purohit, "Unified fractional integral formulae for the generalized Mittag-Leffler functions", J. Sci. Arts, vol. 27, no. 2, pp. 117-124, 2014.

E.T. Wittaker and G.N. Watson, "A course of Modern Analysis", Cambridge: Cambridge Univ. Press, 1962.


DOI: http://dx.doi.org/10.18063/ijmp.v0i0.755
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