Fractional calculus operators and their image formulas

Journal of the Korean Mathematical Society

Volumn 53, Issue 5, 2016, Pages 1183-1210

  Agarwal, Praveen ^a   Choi, Junesang ^b  

^a ANAND INTERNATIONAL COLLEGE OF ENGINEERING   (India)

^b Dongguk University   (South Korea)

Author keywords

Appell s hypergeometric function F3 in two variables; Generalized beta functions of various kinds; Generalized fractional calculus operators; Generalized hypergeometric functions of various kinds; Generalized incomplete hypergeometric functions; Hadamard product; Incomplete gamma functions

Indexed keywords

EID: 84986237286     PISSN: 03049914     EISSN: 22343008     Source Type: Journal    
DOI: 10.4134/JKMS.j150458     Document Type: Article

References (62)

1. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, 1985.
2. P. Agarwal, Further results on fractional calculus of Saigo operators, Appl. Appl. Math. 7 (2012), no. 2, 585-594.
3. P. Agarwal, Generalized fractional integration of the H-function, Matematiche (Catania) 67 (2012), no. 2, 107-118.
4. P. Agarwal, Fractional integration of the product of two multivariables H-function and a general class of polynomials, in: Advances in Applied Mathematics 161 and Approximate Theory, 41, 359-374, Springer Proceedings in Mathematics and Statistics 162, 2013.
5. P. Agarwal, J. Choi, and R. B. Paris, Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl. 8 (2015), no. 5, 451-466.
6. P. Agarwal and S. Jain, Further results on fractional calculus of Srivastava polynomials, Bull. Math. Anal. Appl. 3 (2011), no. 2, 167-174.
7. P. Agarwal, S. Jain, M. Chand, S. K. Dwivedi, and S. Kumar, Bessel functions associated with Saigo-Maeda fractional derivateive operators, J. Fract. Calc. Appl. 5 (2014), no. 2, 9606.
8. F. AI-Musallam and S. L. Kalla, Asymptotic expansions for generalized gamma and incomplete gamma functions, Appl. Anal. 66 (1997), no. 1-2, 173-187.
9. F. AI-Musallam and S. L. Kalla, Further results on a generalized gamma function occurring in diffraction theory, Integral Transforms Spec. Funct. 7 (1998), no. 3-4, 175-190.
10. P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques; Polynômes d’Hermite, Gauthier-Villars, Paris, 1926.
11. M. Caputo, Elasticitae dissipazione Zanichelli, Bologna, 1969.
12. M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, Extension of Euler’s beta function, J. Comput. Appl. Math. 78 (1997), no. 1, 19-32.
13. M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004), no. 2, 589-602.
14. M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, (CRC Press), Boca Raton, London, New York, and Washington, D.C., 2001.
15. J. Choi and P. Agarwal, Certain class of generating functions for the incomplete hyper-geometric functions, Abstr. Appl. Anal. 2014 (2014), Article ID 714560, 5 papes.
16. J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, FILOMAT (2015), accepted for publication.
17. J. Choi and D. Kumar, Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, J. Inequal. Appl. 2014 (2014), Article ID: 499, 15 papes.
18. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Singapore, New York, 2000.
19. S. L. Kalla and R. K. Saxena, Integral operators involving hypergeometric functions, Math. Z. 108 (1969), 231-234.
20. A. A. Kilbas and M. Saigo, Fractional calculus of the H-function, Fukuoka Univ. Sci. Rep. 28 (1998), no. 2, 41-51.
21. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
22. V. S. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res Notes Math. 301, Longman Scientific & Technical; Harlow, Co-published with John Wiley, New York, 1994.
23. V. S. Kiryakova, All the special functions are fractional differintegrals of elementary functions, J. Phys. A 30 (1997), no. 14, 5083-5103
24. V. S. Kiryakova, Multiple (multi-index) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000), no. 1-2, 241-259.
25. V. S. Kiryakova, On two Saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9 (2006), no. 2, 159-176.
26. H. Kober, On fractional integrals and derivatives, Quart. J. Math. Oxford Ser. 11 (1940), 193-212.
27. Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, San Francisco, and London, 1975.
28. M.-J. Luo, G. V. Milovanovic, and P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput. 248 (2014), 631-651.
29. O. I. Marichev, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izv. AN BSSR Ser. Fiz.-Mat. Nauk 1 (1974), 128-129.
30. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Appli-cations, Springer, New York, 2010.
31. A. C. McBride and G. F. Roach (Editors), Fractional Calculus, (University of Strathclyde, Glasgow, Scotland, August 5-11, 1984) Research Notes in Mathematics 138, Pitman Publishing Limited, London, 1985.
32. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
33. K. Nishimoto, Fractional calculus, 1 (1984), 2 (1987), 3 (1989), 4 (1991), 5 (1996), Descartes Press, Koriyama, Japan.
34. K. Nishimoto, An Essence of Nishimoto’s Fractional Calculus, (Calculus of the 21st Century): Integration and Differentiation of Arbitrary Order, Descartes Press, Koriyama, 1991.
35. K. Nishimoto (Editor), Fractional Calculus and Its Applications, (May 29-June 1, 1989), Nihon University (College of Engineering), Koriyama, 1990.
36. K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York, 1974.
37. E. Özergin, Some Properties of Hypergeometric Functions, Ph. D. Thesis, Eastern Mediterranean University, North Cyprus, Turkey, 2011.
38. E. Özergin, M. A. Özarslan, and A. Altın, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), no. 16, 4601-4610.
39. R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania) 69 (2013), no. 2, 33-52.
40. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
41. T. Pohlen, The Hadamard product and universal power series (Dissertation), Universität Trier, 2009.
42. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
43. B. Ross (Editor), Fractional Calculus and Its Applications, (West Haven, Connecticut; June 15-16, 1974), Lecture Notes in Mathematics 457, 1975.
44. M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1977-1978), no. 2, 135-143.
45. M. Saigo, On generalized fractional calculus operators, In: Recent Advances in Applied Mathematics (Proc. Internat. Workshop held at Kuwait Univ.), Kuwait Univ., Kuwait, (1996), 441-450.
46. M. Saigo and A. A. Kilbas, Generalized fractional calculus of the H function, Fukuoka Univ. Sci. Rep. 29 (1999), no. 1, 31-45.
47. M. Saigo and N. Maeda, More generalization of fractional calculus, In: Transform methods & special functions, Varna ’96, 386-400, Bulgarian Acad. Sci., Sofia, 1998.
48. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon et alibi, 1993.
49. R. K. Saxena and M. Saigo, Generalized fractional calculus of the H-function associated with the Appell function, J. Frac. Calc. 19 (2001), 89-104.
50. R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russ. J. Math. Phys. 20 (2013), no. 1, 121-128.
51. R. Srivastava, Some generalizations of Pochhammer’s symbol and their associated families of hypergeometric functions and hypergeometric polynomials, Appl. Math. Inf. Sci. 7 (2013), no. 6, 2195-2206.
52. R. Srivastava and N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput. 219 (2012), no. 6, 3219-3225.
53. R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their applications involving generalized hypergeometric polynomials, Appl. Math. Comput. 234 (2014), 277-285.
54. H. M. Srivastava and P. Agarwal, Certain fractional integral operators and the general-ized incomplete hypergeometric functions, Appl. Appl. Math. 8 (2013), no. 2, 333-345.
55. H. M. Srivastava, A. C¸ etinkaya, and ˙ I. O. Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 (2014), 484-491.
56. H. M. Srivastava, M. A. Chaudhry, and R. P. Agarwal, The incomplete Pochhammer symbols and their applications to hypergeometric and related functions, Integral Transforms Spec. Funct. 23 (2012), no. 9, 659-683.
57. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
58. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
59. H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001), no. 1, 1-52.
60. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathe-matical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1996.
61. F. G. Tricomi, Sulla funzione gamma incompleta, Ann. Mat. Pura Appl. (4) 31 (1950), 263-279.
62. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions.Reprint of the fourth (1927) edition.Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996.