FURTHER EXTENSION OF EXTENDED FRACTIONAL DERIVATIVE OPERATOR OF RIEMANN-LIOUVILLE

The main objective of this present paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. Also, we give some results related to the newly defined fractional operator such as Mellin transform and relations to extended hypergeometric and Appell’s function via generating functions.


introduction
Recently, the application and importance of fractional calculus have been paid more attention.In the field of mathematical analysis, the fractional calculus is a more helpful tool to find out differentials and integrals with the real numbers or with the complex numbers powers of the fractional calculus.Various extensions and generalization of fractional derivative operators are recently investigated by the researchers (see [6,8,12,16,17]).
For various extensions and generalization the readers may follow the recent work of researchers (see e.g., [1,4,9]).Parmar et al. [14] introduced the following extended beta function as where K v+ 1 2 (.) is the modified Bessel function of order v + 1 2 .Clearly, when v = 0 then (1.15) reduces to (1.8) by using the fact that K1 2 z = π 2z e −z .Also, the following extended hypergeometric and confluent hypergeometric functions and their integral representation respectively as (see, [14]): ) They also obtained the following transformation formula for extended confluent hypergeometric function It is clear that, when v = 0 then the equations (1.16)-(1.19)reduce to the extended hypergeometric, confluent hypergeometric functions and their integral representations defined in (1.9)-(1.12)respectively by using the fact that Recently Dar and Paris [5] have introduced the following Appell's hypergeometric function by where In the same paper, they [5] defined its integral representation as: where (p Obviously, when v = 0 in (1.21) and (1.22) then we get the extended Appell function and its integral representation (see , (1.13) and (1.14)) by using the fact that Similarly, when v = p = 0 then (1.21) and (1.22) reduce to the well-known classical Appell's function and its integral representation.

Extension of fractional derivative operator
In this section, we define further extension of extended Riemann-Liouville fractional derivative.
Theorem 2.2.Let (µ) > 0 and suppose that the function f (z) is analytic at the origin with its Maclaurin expansion given by f (z) = ∞ n=0 a n z n where |z| < δ for some δ ∈ R + .Then Proof.Using the series expansion of the function f (z) in (2.7) gives As the series is uniformly convergent on any closed disk centered at the origin with its radius smaller then δ, therefore the series so does on the line segment from 0 to a fixed z for |z| < δ.Thus it guarantee terms by terms integration as follows which is the required proof.
Proof.By direct calculation, we have dt.
Substituting t = zu in the above equation, we get Using (1.16) and after simplification we get the required proof.
Proof.Using the power series for (1 − z) −α and applying Theorem 2.5 with η = n, we can write which is the required proof.

Generating relations and some further results
In this section, we derive some generating relations of linear and bilinear type for the extended (p, v)-hypergeometric functions.
Proof.Consider the following series identity Thus, the power series expansion yields Multiplying both sides of (3.2) by z β− 3 2 and then applying the operator D β−γ;λ,ρ z;p,q on both sides, we have Interchanging the order of summation and the operator D β−γ z;p,v , we have Thus by applying Theorem 2.3, we obtain the required result.
Proof.Consider the series identity Using the power series expansion to the left sides, we have Multiplying both sides of (3.4) by z α− 3 2 (1 − z) −δ and applying the operator D α−γ z;p,v on both sides, we have , where (α) > 0 and |zt| < |1 − t|, thus by Theorem 2.2, we have Applying Theorem 2.4 on both sides, we get the desired result.

Concluding remarks
In this paper, we established the extension of extended fractional derivative operator.We conclude that when v = 0 and using the fact that K 1 2 z = π 2z e −z then all the results established in this paper will reduce to the results obtained by Kiymaz et al. see [6].Also, when p = v = 0 then we get the results related to the classical Reimann-Liouville fractional derivative operator.