Extended Riemann-Liouville type fractional derivative operator with applications

Abstract The main purpose of this paper is to introduce a class of new extended forms of the beta function, Gauss hypergeometric function and Appell-Lauricella hypergeometric functions by means of the modified Bessel function of the third kind. Some typical generating relations for these extended hypergeometric functions are obtained by defining the extension of the Riemann-Liouville fractional derivative operator. Their connections with elementary functions and Fox’s H-function are also presented.


Introduction
Extensions and generalizations of some known special functions are important both from the theoretical and applied point of view. Also many extensions of fractional derivative operators have been developed and applied by many authors (see [2-6, 11, 12, 19-21] and [17,18]). These new extensions have proved to be very useful in various elds such as physics, engineering, statistics, actuarial sciences, economics, nance, survival analysis, life testing and telecommunications. The above-mentioned applications have largely motivated our present study.
The extended incomplete gamma functions constructed by using the exponential function are de ned by and with arg z < π, which have been studied in detail by Chaudhry and Zubair (see, for example, [2] and [4]). The extended incomplete gamma functions γ (α, z; p) and Γ (α, z; p) satisfy the following decomposition formula (α, z; p) + Γ (α, z; p) = Γ p (α) = p α K α √ p (R(p) > ) , where Γ p (α) is called extended gamma function, and K α (z) is the modi ed Bessel function of the third kind, or the Macdonald function with its integral representation given by (see [7]) where R(z) > and For α = , we have Instead of using the exponential function, Chaudhry and Zubair extended (1) and (2) in the following form (see [3], see also [4]) and where Inspired by their construction of (7) and (8), we aim to introduce a class of new special functions and fractional derivative operator by suitably using the modi ed Bessel function K α (z).
The present paper is organized as follows: In Section 2, we rst de ne the extended beta function and study some of its properties such as di erent integral representations and its Mellin transform. Then some extended hypergeometric functions are introduced by using the extended beta function. The extended Riemann-Liouville type fractional derivative operator and its properties are given in Section 3. In Section 4, the linear and bilinear generating relations for the extended hypergeometric functions are derived. Finally, the Mellin transforms of the extended fractional derivative operator are determined in Section 5.

Extended beta and hypergeometric functions
This section is divided into two subsections. In subsection-1, we de ne the extended beta function B µ (x, y; p; m) and study some of its properties. In subsection-2, we introduce the extended Gauss hypergeometric function F µ (a, b; c; z; p; m), the Appell hypergeometric functions F ,µ , F ,µ and the Lauricella hypergeometric function F D ,µ and then obtain their integral representations. Throughout the present study, we shall assume that R(p) > and m > .

. Extended beta function
De nition 2.1. The extended beta function B µ (x, y; p; m) with R(p) > is de ned by where x, y ∈ C, m > and R(µ) ≥ .
Theorem 2.3. The following integral representations for the extended beta functions B µ (x, y; p; m) with R(p) > are valid Proof. These formulas can be obtained by using the transformations t = cos θ, t = u +u and t = +u in (9), respectively.

Theorem 2.4. The following expression holds true
where κ (p u) is given by (5).
Proof. Expressing B µ (x, y; p, m) in its integral form with the help of (9), and taking (4) into account, we obtain where κ (p u) is given by (5).
In order to write the inner integral as our extended beta function, we need the following variant of (6), that is, Then, we have Substituting (16) into (15) we obtain the required result (14).

Remark 2.5.
It is interesting to note that using the de nition of the extended beta function (see, [11,12]) we can get the following expression for (9) where the function B b;ρ,λ (x, y) is given by [12, p. 631, Eq. (2)] The following theorem establishes the relation between the Mellin transform and the extended beta function.
Then we have the following relation Proof. First, we have Since the Mellin transform of the Macdonald function K v (z) is given by [9, p. 37, Eq.(1.7.41)]: the last integral in (19) can be evaluated as where we have used Finally, we get Now we can derive the Fox H-function representation of the extended beta function de ned in (9). Let m, n, p, q be integers such that ≤ m ≤ q, ≤ n ≤ p, and for parameters a i , b i ∈ C and for parameters where with the contour L suitably chosen. As convention, the empty product is equal to one. The theory of the Hfunction is well explained in the books of Mathai [14], Mathai (18) and (20) mean H m,n p,q (z) in (22) is the inverse Mellin transform of Θ (s) in (23).
where B µ (x, y; p; m) is as de ned in (9).
Here, it is important to mention that when we take m = , µ = and then letting p → , function (24) reduces to the ordinary Gauss hypergeometric function de ned by where (x) n denotes the Pochhammer symbol de ned, in terms of the familiar gamma function, by For conditions of convergence and other related details of this function, see [1], [9] and [16]. Similarly, we can reduce the functions (25), (26) and (27) to the well-known Appell functions F , F and Lauricella function F D , respectively (see [16] and [23]). Now, we establish the integral representations of the extended hypergeometric functions given by (24), (25), (26) and (27) as follows.
Proof. By using (9) and employing the binomial expansion we get the above integral representation.
Theorem 2.13. The following integral representation for the extended hypergeometric function F ,µ is valid Proof. For simplicity, let I denote the left-hand side of (31). Then, using (25) yields By applying (9) to the integrand of (31), after a little simpli cation, we have B(a, d − a) x n n! y k k! .
By interchanging the order of summation and integration in (33), we get which proves the integral representation (31).
To establish Theorem 2.13, we need to recall the following elementary series identity involving the bounded sequence of {f (N)} ∞ N= stated in the following result.
Proof. Let L denote the left-hand side of (36). Then, using (26) yields By applying (9) to the integrand of (32), we have Next, interchanging the order of summation and integration in (38), which is guaranteed, yields Finally, applying (35) to the double series in (39), we obtain the right-hand side of (36).
Theorem 2.16. The following integral representation for the extended hypergeometric function F D,µ is valid Proof. A similar argument in the proof of Theorem 2.15 will be able to establish the integral representation in (40). Therefore, details of the proof are omitted.

Extended Riemann-Liouville fractional derivative operator
We rst recall that the classical Riemann-Liouville fractional derivative is de ned by (see [23, p. 286]) where R(ν) < and the integration path is a line from to z in the complex t-plane. It coincides with the fractional integral of order −ν. In case m − < R (ν) < m, m ∈ N, it is customary to write We present the following new extended Riemann-Liouville-type fractional derivative operator.
For n − < R (ν) < n, n ∈ N, we write Proof. Using De nition 3.1 and 1, we have Next, we apply the extended Riemann-Liouville fractional derivative to a function f (z) analytic at the origin. Since the power series converges uniformly on any closed disk centered at the origin with its radius smaller than ρ, so does the series on the line segment from to a xed z for z < ρ. This fact guarantees term-by-term integration as follows: As a consequence we have the following result.