The purpose of this section is to develop two representations of the generalized fractional derivative operator in terms of the common Riemann-Liouville fractional operator (1). These representations are found by making use of a new transformation formula for fractional derivatives published recently by the authors [16]. Some examples of possible new relationships are also given.
Note that for the remainder of this paper, will denote the Pochhammer’s symbol defined by
(8)
Moreover, we adopt the following notation to denote the generalized hypergeometric function
(9)
The first representation for the fractional derivative operator is contained in the next theorem.
Theorem 1
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(i)
Let ℛ be a simply connected region containing the origin.
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(ii)
Let be analytic in ℛ. Then, for with , the following relation holds true
(10)
Proof Consider the integral representation of the operator in the complex plane with :
(11)
Making the following change of variables (), we have
(12)
Note that we must have in the right side of (12) after the evaluation of the fractional derivative. So, the point w must be near the point z. Using this fact, we can expand the expression in power series. We thus have
(13)
Rewriting the integral in (13) in terms of the Riemann-Liouville fractional derivative operator yields the desired result. □
Corollary 2 Substituting in Theorem 1, we obtain
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Example 1 Let in Theorem 1. We obtain for the l.h.s. of (10) by making use of the power series expansion of
(15)
For the fractional derivative operator involved in the r.h.s. of (10), we have
Combining (15) and (16) in (10), we obtain after some simple calculations
where denotes the Gauss hypergeometric function [24].
Moreover, setting in (17) gives the following relationship involving the Gauss hypergeometric function:
(18)
Recently, Tremblay et al. [16] discovered a new transformation formula for the fractional derivatives. Many interesting applications of this formula has also been given. Especially, they proved the next result.
Theorem 3 Let be an analytic function in the simply connected region ℛ containing the origin. For , we have
(19)
Note that we must have in the right side of (19) after the evaluation of the fractional derivative since the point w must be near the point z.
With the help of this new result, we can easily obtain the next theorem.
Theorem 4
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(i)
Let ℛ be a simply connected region containing the origin.
-
(ii)
Let be analytic in ℛ. Then, for with , the following relation holds true
(20)
Proof Applying Theorem 3 to Theorem 1, the result follows easily. □
Corollary 5 Substituting in the Theorem 4, we obtain
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Example 2 Putting in Theorem 4 gives for the l.h.s. of (20)
(22)
Now, for the r.h.s. of (20), the computation of the fractional derivative operator applied to the exponential function yields
(23)
Replacing (22) and (23) in (20), we arrive, after some simplifications, to the next formula:
(24)
Note that by making use of Theorem 1, we also have that