Abstract
In this chapter, our aim is to establish certain new image formulae of generalized hypergeometric functions by using the operators of fractional calculus. Some new image formulae are obtained by applying specific integral transforms on resulting image formulae. We also acquired generalization of fractional kinetic equations involving extended hypergeometric functions.
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References
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies (Elsevier Science, Amsterdam, 2006)
A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function Theory and Applications (Springer, New York, 2010)
A. Wiman, Uber de fundamental theorie der funktionen \(E_{\alpha }(x)\). Acta Mathematica 29(1), 191–201 (1905)
D. Kumar, Solution of fractional kinetic equation by a class of integral transform of pathway type. J. Math. Phys. 54, 043509 (2013). doi:10.1063/1.4800768
D. Kumar, S.D. Purohit, A. Secer, A. Atangana, On generalized fractional kinetic equation involving generalized Bessel functions of the first kind. Math. Probl. Eng. 2015, Article ID 289387, 7 pp. (2015)
E.D. Rainville, Special Functions (Macmillan Company, New York, 1960) (Reprinted by Chelsea Publishing Company, Bronx, 1971)
H.J. Haubold, A.M. Mathai, The fractional kinetic equation and thermonuclear functions. Astrophys. Space Sci. 273(1), 53–63 (2000)
H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester) (Wiley, New York, 1984)
H.M. Srivastava, J. Choi, Zeta and \(q\) -Zeta Functions and Associated Series and Integrals (Elsevier Science, Amsterdam, 2012)
H.M. Srivastava, M. Saigo, Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation. J. Math. Anal. Appl. 121, 325–369 (1987)
H.M. Srivastava, P. Agarwal, S. Jain, Generating functions for the generalized Gauss hypergeometric functions. Appl. Math. Comput. 247, 348–352 (2014)
H.M. Srivastava, P.W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester) (Wiley, New York, 1985)
H.M. Srivastava, R. Agarwal, S. Jain, Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions. Math. Methods Appl. Sci. (Article in press)
H.M. Srivastava, R.K. Saxena, Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)
I.J. Podulbuny, Fractional Differential Equations (Academic Press, New York, 1999)
I.N. Sneddon, The Use of Integral Transform (Tata McGraw-Hill, New Delhi, 1979)
I.S. Jeses, J.A.T. Machado, Fractional control of heat diffusion systems. Nonlinear Dyn. 54(3), 263–282 (2008)
J. Choi, D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions. Math. Commun. 20, 113–123 (2015)
J.C. Prajapati, K.B. Kachhia, Fractional modeling of temperature distribution and heat flux in the semi infinite solid. J. Fract. Calc. Appl. 5(2), 38–43 (2014)
J.C. Prajapati, K.B. Kachhia, S.P. Kosta, Fractional calculus approach to study temperature distribution within a spinning satellite. Alex. Eng. J. 55, 2345–2350 (2016)
K.B. Kachhia, J.C. Prajapati, Solution of fractional partial differential equation aries in study of heat transfer through diathermanous material. J. Interdiscip. Math. 18(1–2), 125–132 (2015)
K.B. Kachhia, J.C. Prajapati, On generalized fractional kinetic equations involving generalized Lommel-Wright functions. Alex. Eng. J. 55, 2953–2957 (2016)
K.B. Kachhia, J. Choi, J.C. Prajapati, S.D. Purohit, Some integral transforms involving extended generalized Gauss hypergeometric functions. Commun. Korean Math. Soc. 31(4), 779–790 (2016)
M.J. Luo, G.V. Milovanovic, P. Agarwal, Some results on the extended beta and extended hypergeometric functions. Appl. Math. Comput. 248, 631–651 (2014)
M.J. Luo, R.K. Raina, On certain classes of fractional kinetic equations. Filomat 28(10), 2077–2090 (2014)
M. Saigo, A remark on integrals operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135–143 (1978)
M. Saigo, N. Maeda, More generalization of fractional calculus, in Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, Verna, 1996, IMI-BAS. Sofia (1998), pp. 386–400
P. Agarwal, Certain properties of the generalized Gauss hypergeometric functions. Appl. Math. Inf. Sci. 8(5), 2315–2320 (2014)
P. Agarwal, J. Choi, Fractional calculus operators and their image formulae. J. Korean Math. Soc. 53(5), 1183–1210 (2016)
P. Agarwal, J. Choi, K.B. Kachhia, J.C. Prajapati, H. Zhou, Some integral transforms and fractional integral formulas for the extended hypergeometric functions. Commun. Korean Math. Soc. 31(3), 591–601 (2016)
P. Agarwal, M. Chand, E.T. Karimov, Certain image formulae of generalized hypergeometric functions. Appl. Math. Comput. 266, 763–772 (2015)
P. Agarwal, M. Chand, G. Singh, Certain fractional kinetic equations involving the product of generalized k-Bessel function. Alex. Eng. J. (Article in press)
P. Appell, J. Kamp\(\acute{e}\) de F\(\acute{e}\)riet, Fonctions Hyperg \(\acute{e}\) triques et Hypersph \(\acute{e}\) riques, Polyn \(\hat{o}\) mes d’Hermite (Gauthier-Villars, Paris, 1926)
R. Agarwal, S. Jain, R.P. Agarwal, Solution of fractional volterra integral equation and non-homogeneous time fractional heat equation using integral transform of pathway type. Prog. Fract. Differ. Appl. 1(3), 145–155 (2015)
R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)
R.K. Saxena, A.M. Mathai, H.J. Haubold, On generalized fractional kinetic equation. Phys. A 344, 657–664 (2004)
R.K. Saxena, M. Saigo, Generalised fractional calculus of the H-function associated with the Appell function. J. Fract. Calc. 19, 89–104 (2001)
R.K. Saxena, S.L. Kalla, On the solution of certain fractional kinetic equations. Appl. Math. Comput. 199, 504–511 (2008)
S.G. Samko, A. Kilbas, O. Marichev, Fractional Integral and Derivatives. Theory and Applications (Gordon and Breach Science Publishers, New York, 1990)
T. Pohlen, The Hadamard product and universal power series. Dissertation, Universität Trier (2009)
V.B.L. Chaurasia, S.C. Pandey, On the new computable solution of the generalized fractional kinetic equations involving the generalized function for fractional calculus and related functions. Astrophys. Space Sci. 317, 213–219 (2008)
V.B.L. Chaurasia, S.C. Pandey, Computable extensions of generalized fractional kinetic equations in astrophysics. Res. Astron. Astrophys. 10(1), 22–32 (2010)
V. Kourganoff, Introduction to the Physics of Stellar Interiors (D. Reidel Publishing Company, Dordrecht, 1973)
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Kachhia, K.B., Agarwal, P., Prajapati, J.C. (2017). Certain Image Formulae and Fractional Kinetic Equations Involving Extended Hypergeometric Functions. In: Ruzhansky, M., Cho, Y., Agarwal, P., Area, I. (eds) Advances in Real and Complex Analysis with Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-10-4337-6_1
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