Keywords: RiemannLiouville type fractional integral operator, Sfunction, Kampe de Feriet function, Fox’s H function, (2000 Mathematics subject classification: 33C99)
Applied Mathematics and Physics, 2013 1 (2),
pp 2730.
DOI: 10.12691/amp122
Received June 25, 2013; Revised August 14, 2013; Accepted August 15, 2013
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
Thefunction occurring in the paper will be defined and represented as follows:
 (1.1) 
Where
 (1.2) 
Which contains fractional powers of the gamma functions. Here, and throughout the paper and are complex parameters, (not all zero simultaneously) and exponents and can take on non integer values.
The following sufficient condition for the absolute convergence of the defining integral for the function given by equation (1.1) have been given by (Buschman and Srivastava ^{[1]}).
 (1.3) 
 (1.4) 
The behavior of the function for small values of follows easily from a result recently given by (Rathie ^{[4]}, eq.(6.9)).
We have
 (1.5) 
If we take in (1.1), the function reduces to the Fox’s Hfunction ^{[3]}.
We shall use the following notation:
Srivastava and Daoust ^{[6]} have generalized the Kampe de Feriet function as function which is defined and represented as^{[8]}:
 (1.6) 
The Beta function is defined as:
 (1.7) 
2. Preliminary Result
 (2.1) 
Where,
 (2.2) 
Provided
(i)
(ii)
(iii)
Where is given by (1.3).
Proof: Express the  function on the left hand side of (2.1) as a series uses (1.6) and express the function as a contour integral using (1.1) and then change the order of integration and summation and evaluate the inner integral using (1.7) to get the required result. The change of order of integration and summation is justified, when the given conditions are satisfied because of the absolute convergence of the integrals involved.
3. Main Result
The Integral Equation
(i), has the solution
(ii)
 (3.1) 
Where
and is given by (2.2), provided:
(i)
(ii)
(iii)
Where is given by (1.3).
(iv) and
(v)is continuous in .
Proof: Substituting for from (ii) of (3.1) and changing the order of integration the left hand side of (i) of (3.1) becomes
 (3.2) 
On putting , the inner integral in (3.2) becomes:
Which on using (2.1), reduces to .
Now (3.2) becomes:
4. Special Cases:
When , the function reduces to the Fox’s function and (3.1) reduces to the result:
The integral equation
(i), has the solution
(ii)
 (4.1) 
Where
and is given by (2.2), provided:
(i)
(ii)
(iii)
Where
(iv) and
(v)is continuous in .
When , the function reduces to the Meijer’sfunction and (4.1) reduces to the result:
The integral equation
(i), has the solution
(ii)
 (4.2) 
Where
and is given by (2.2), provided:
(i)
(ii)
(iii)
Where
(iv) and
(v)is continuous in .
5. Application
We shall define the RiemannLiouville fractional derivative of function of order (or alternatively, order fractional integral) ^{[2, 7]} by
 (5.1) 
Where is a positive integer and the integral exists. For simplicity the special case of the fractional derivative operator, when will be written as . Thus we have
 (5.2) 
The preliminary result (2.1) can be rewritten as the following fractional integral formula:
 (5.3) 
Where and all the conditions of validity mentioned with (2.1) are satisfied.
6. Conclusion
Thus, our obtained result the fractional integral formula given by (5.3) is also quite general in nature and can easily yield RiemannLiouville fractional integrals of a large number of simpler functions and polynomials merely by specializing the parameters occurring in it which may find applications in electromagnetic theory and probability.
References
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 In article  

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