Browse > Article
http://dx.doi.org/10.4134/JKMS.j150458

FRACTIONAL CALCULUS OPERATORS AND THEIR IMAGE FORMULAS  

Agarwal, Praveen (Department of Mathematics and International College of Engineering)
Choi, Junesang (Department of Mathematics Dongguk University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.5, 2016 , pp. 1183-1210 More about this Journal
Abstract
During the past four decades or so, due mainly to a wide range of applications from natural sciences to social sciences, the so-called fractional calculus has attracted an enormous attention of a large number of researchers. Many fractional calculus operators, especially, involving various special functions, have been extensively investigated and widely applied. Here, in this paper, in a systematic manner, we aim to establish certain image formulas of various fractional integral operators involving diverse types of generalized hypergeometric functions, which are mainly expressed in terms of Hadamard product. Some interesting special cases of our main results are also considered and relevant connections of some results presented here with those earlier ones are also pointed out.
Keywords
generalized fractional calculus operators; generalized beta functions of various kinds; generalized hypergeometric functions of various kinds; Hadamard product; incomplete gamma functions; generalized incomplete hypergeometric functions; Appell's hypergeometric function $F_3$ in two variables;
Citations & Related Records
연도 인용수 순위
  • Reference
1 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.
2 R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russ. J. Math. Phys. 20 (2013), no. 1, 121-128.   DOI
3 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.   DOI
4 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.
5 R. Srivastava and N. E. Cho, Some extended Pochhammer symbols and their applications involving general- ized hypergeometric polynomials, Appl. Math. Comput. 234 (2014), 277-285.
6 H. M. Srivastava and P. Agarwal, Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math. 8 (2013), no. 2, 333-345.
7 H. M. Srivastava, A. Cetinkaya, and I. O. Kiymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 (2014), 484-491.
8 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.   DOI
9 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.
10 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.
11 H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001), no. 1, 1-52.
12 N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1996.
13 F. G. Tricomi, Sulla funzione gamma incompleta, Ann. Mat. Pura Appl. (4) 31 (1950), 263-279.   DOI
14 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.
15 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.
16 L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, 1985.
17 P. Agarwal, Further results on fractional calculus of Saigo operators, Appl. Appl. Math. 7 (2012), no. 2, 585-594.
18 P. Agarwal, Generalized fractional integration of the $\overline{H}$-function, Matematiche (Catania) 67 (2012), no. 2, 107-118.
19 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.   DOI
20 P. Agarwal and S. Jain, Further results on fractional calculus of Srivastava polynomials, Bull. Math. Anal. Appl. 3 (2011), no. 2, 167-174.
21 P. Appell and J. Kampe de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier-Villars, Paris, 1926.
22 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.
23 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.   DOI
24 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.   DOI
25 M. Caputo, Elasticitae dissipazione Zanichelli, Bologna, 1969.
26 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.   DOI
27 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.
28 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.
29 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.
30 J. Choi and P. Agarwal, A note on fractional integral operator associated with multiindex Mittag-Leffler functions, FILOMAT (2015), accepted for publication.
31 A. A. Kilbas and M. Saigo, Fractional calculus of the H-function, Fukuoka Univ. Sci. Rep. 28 (1998), no. 2, 41-51.
32 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.
33 R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Singapore , New York, 2000.
34 S. L. Kalla and R. K. Saxena, Integral operators involving hypergeometric functions, Math. Z. 108 (1969), 231-234.   DOI
35 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.
36 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.
37 V. S. Kiryakova, All the special functions are fractional differintegrals of elementary functions, J. Phys. A 30 (1997), no. 14, 5083-5103
38 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.   DOI
39 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.
40 H. Kober, On fractional integrals and derivatives, Quart. J. Math. Oxford Ser. 11 (1940), 193-212.
41 Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, San Francisco, and London, 1975.
42 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.
43 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.
44 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.
45 A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010.
46 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
47 K. Nishimoto, Fractional calculus, 1 (1984), 2 (1987), 3 (1989), 4 (1991), 5 (1996), Descartes Press, Koriyama, Japan.
48 K. Nishimoto, An Essence of Nishimoto's Fractional Calculus, (Calculus of the 21st Century): Integration and Differentiation of Arbitrary Order, Descartes Press, Koriyama, 1991.
49 K. Nishimoto (Editor), Fractional Calculus and Its Applications, (May 29-June 1, 1989), Nihon University (College of Engineering), Koriyama, 1990.
50 K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, New York, 1974.
51 E. Ozergin, Some Properties of Hypergeometric Functions, Ph. D. Thesis, Eastern Mediterranean University, North Cyprus, Turkey, 2011.
52 T. Pohlen, The Hadamard product and universal power series (Dissertation), Universitat Trier, 2009.
53 E. Ozergin, M. A. Ozarslan, and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), no. 16, 4601-4610.   DOI
54 R. K. Parmar, A new generalization of Gamma, Beta, hypergeometric and confluent hypergeometric functions, Matematiche (Catania) 69 (2013), no. 2, 33-52.
55 I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
56 E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
57 B. Ross (Editor), Fractional Calculus and Its Applications, (West Haven, Connecticut; June 15-16, 1974), Lecture Notes in Mathematics 457, 1975.
58 M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1977/78), no. 2, 135-143.
59 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.
60 M. Saigo and A. A. Kilbas, Generalized fractional calculus of the H function, Fukuoka Univ. Sci. Rep. 29 (1999), no. 1, 31-45.
61 M. Saigo and N. Maeda, More generalization of fractional calculus, In: Transform methods & special functions, Varna '96, 386-400, Bulgarian Acad. Sci., Sofia, 1998.
62 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.