FRACTIONAL DERIVATIVE PERTAINING TO I-FUNCTION AND LAURICELLA FUNCTION

*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:

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  2. B.L. Chaurasia, fractional derivative of the multivariable polynomials, Bull. Malaysian Math. Sc. Soc.(second series), 26 (2003),1-8
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