*HARSHITA GARG AND **ASHOK SINGH SHEKHAWAT
*Suresh Gyan Vihar University, Jagatpura, Jaipur, Rajasthan, India
**Arya College Of Engineering And Information Technology, Jaipur, Rajasthan, India
Abstract :
The object of this paper is to obtain a fractional derivative of I- function associated with generalize Lauricella functions and general class of multivariable polynomials.
Key words: Fractional derivative operator, I-function, Lauricella function, general class of multivariable polynomials.
INTRODUCTION:
The I- function given by Saxena [4] is represented and defined as following:
pi (i= 1,….,R), qi (i= 1,….,R), e, f are integers satisfying 0 £ f £ pi, 0 £ e £ qi (i= 1,….,R); R is finite, αj, βj, αji, βji, are real and positive; aj, bj, aji, bji are complex numbers and £ is the path of integration separating the increasing and decreasing sequences of poles of the integrand.
The integral converges if
,
Now following shorthand notations given by Srivastava and Daoust [6] denote the generalized Lauricella function of several complex variables.
= F … (1.9)
Proof: In order to prove (2.1), express the I-function in terms of Mellin-Barnes type of contour integrals by (1.1) and Lauricella function by (1.7) and general class of polynomials given by (1.4), then collecting the powers of (x-u) and (v-x). Finally making use of the result (1.6), we get the main result (2.1).
PARTICULAR CASES:
If we take R=1 in (1.1), then I-function breaks into well known Fox’s H-function and consequently there hold the following result:
Valid under the conditions surrounding (2.1)
- If we take multivariable H-function in place of I-function in (2.1), then we have a known result obtained by Chaurasia and Singhal [2] as following:
Valid under the conditions surrounding (2.1)
Taking R=1 and replacing f→f1,….,fr in (2.1),
we get a known result obtained by Chaurasia and Shekhawat [1].
Conclusion:
The main result derived here is of a very general Nature and hence encompass several cases of interest hitherto scattered in the literature.
REFERENCES:
- B.L. Chaurasia and A.S. Shekhawat, fractional derivative associated with the multivariable polynomials, Kyungpook math. J. 47 (2007),495-500.
- B.L. Chaurasia, fractional derivative of the multivariable polynomials, Bull. Malaysian Math. Sc. Soc.(second series), 26 (2003),1-8
- B. Oldham and J. Spanier, The fractional calculus, Academic Press New York, 1974.
- P.Saxena, The I-function, Anamaya Publishers, New Delhi (2008)
- Srivastava, H.M., A multilinear generating function for the Konhauser sets of bi-orthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 117, 183-191 (1985)
- M. Srivastava, and M.C. Daoust, certain generalized neuman expansions associated with the kempe’ de f’eriet function, Nederl.Akad.Wetensch Indag.Math., 31 (1969), 449-457.
- M. Srivastava, and M. Garg, some integrals involving a general class of polynomials and the multivariable H-function, Rev. Roumanie phys.,32 (1987), 685-692
- M. Srivastava, and S.P.Goyal, fractional derivatives of the H-function of several variables, J.Math. Anal. Appl. 112 (1985), 641-651.
- M. Srivastava, and R. Panda, certain expansion formulas involving the generalized Lauricella function, II comment math. Univ. St. Paul, 24 (1974), 7-14