FRACTIONAL DIFFERENTIATION OF THE PRODUCT OF APPELL FUNCTION F_{3} AND MULTIVARIABLE H-FUNCTIONS

- Journal title : Communications of the Korean Mathematical Society
- Volume 31, Issue 1, 2016, pp.115-129
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2016.31.1.115

Title & Authors

FRACTIONAL DIFFERENTIATION OF THE PRODUCT OF APPELL FUNCTION F_{3} AND MULTIVARIABLE H-FUNCTIONS

Choi, Junesang; Daiya, Jitendra; Kumar, Dinesh; Saxena, Ram Kishore;

Choi, Junesang; Daiya, Jitendra; Kumar, Dinesh; Saxena, Ram Kishore;

Abstract

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. [9] evaluated the fractional integrals of the product of Appell function of the third kernel and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, -Kober and Saigo operators, associated with multivariable H-function and Appell function are shown to be deduced as special cases of our findings.

Keywords

multivariable H-function;Saigo fractional calculus operators;Saigo-Maeda operators;fractional calculus;Appell function ;H-function;Riemann-Liouville derivative operator;

Language

English

References

1.

A. A. Kilbas, Fractional calculus of the generalized Wright function, Fract. Calc. Appl. Anal. 8 (2005), no. 2, 113-126.

2.

A. A. Kilbas and M. Saigo, Fractional calculus of the H-function, Fukuoka Univ. Sci. Rep. 28 (1998), no. 2, 41-51.

3.

A. A. Kilbas and N. Sebastian, Generalized fractional differentiation of Bessel function of the first kind, Math. Balkanica (N. S.) 22 (2008), no. 3-4, 323-346.

4.

D. Kumar and J. Daiya, Generalized fractional differentiation of the ${\bar{H}}$ -function involving general class of polynomials, Int. J. Pure Appl. Sci. Technol. 16 (2013), no. 2, 42-53.

5.

A. M. Mathai and R. K. Saxena, The H-Function with Application in Statistics and Other Disciplines, John Wiley & Sons. Inc., 1978.

6.

A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function, Springer, New York, 2010.

7.

M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1978), no. 2, 135-143.

8.

M. Saigo and A. A. Kilbas, Generalized fractional calculus of the H-function, Fukuoka Univ. Sci. Rep. 29 (1998), 31-45.

9.

M. Saigo and N. Maeda, More generalization of fractional calculus, Tran. Meth. Spec. Fun., Varna, Bulgaria, (1996), 386-400.

10.

M. Saigo, R. K. Saxena, and J. Ram, Fractional integration of the product of Appell function $F_3$ and multivariable H-function, J. Fract. Calc. 27 (2005), 31-42.

11.

S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, New York, 1993.

12.

R. K. Saxena, J. Ram, and J. Daiya, Fractional integration of the multivariable H-function VIA pathway operator, Ganita Sandesh 25 (2011), no. 1, 1-10.

13.

R. K. Saxena, J. Ram, J. Daiya, and T. K Pogany, Inequalities associated with Cebysev functional for Saigo fractional integration operator, Integral Transforms Spec. Funct. 22 (2011), no. 9, 671-680.

14.

R. K. Saxena, J. Ram, and D. Kumar, Generalized fractional differentiation of the Aleph-Function associated with the Appell function $F_3$ , Acta Ciencia Indica 38 M (2012), no. 4, 781-792.

15.

R. K. Saxena, J. Ram, and D. Kumar, Generalized Fractional Integration of the Product of two ${\aleph}$ -Functions Associated with the Appell Function $F_3$ , Romai J. 9 (2013), no. 1, 147-158.

16.

R. K. Saxena and M. Saigo, Generalized fractional calculus of the H-function associated with Appell function $F_3$ , J. Fract. Calc. 19 (2001), 89-104.

17.

H. M. Srivastava and M. Garg, Some integrals involving a general class of polynomials and the multivariable H-function, Rev. Roumaine Phys. 32 (1987), no. 7, 685-692.

18.

H. M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. 283/284 (1976), 265-274.

19.

H. M. Srivastava and R. Panda, Some expansion theorems and the generating relations for the H-function of several complex variables. II, Comment. Math. Univ. St. Paul 25 (1976), no. 2, 167-197.

20.

H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001), no. 1, 1-52.