ON GENERALIZED k-FRACTIONAL DERIVATIVE OPERATOR

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to khypergeometric and k-Appell’s functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.


introduction
The classical beta function and its relation with well known gamma function is given by The Gauss hypergeometric, confluent hypergeometric and Appell's functions which are respectively defined by(see [14]) ( δ 1 , δ 2 , δ 3 ∈ C and δ 3 ̸ = 0, −1, −2, −3, • • • ) , and , respectively.The Appell's series or bivariate hypergeometric series is defined by The integral representation of hypergeometric, confluent hypergeometric and Appell's functions are respectively defined by ) , and (1.7) The k-gamma function, k-beta function and the k-Pochhammer symbol introduced and studied by Diaz and Pariguan [1].The integral representation of k-gamma function and k-beta function respectively given by (1.9) Here, we recall the following relations (see [1]).
In 2015, Mubeen et al. [11] introduced k-Appell hypergeometric function as k and k > 0. Also, they define its integral representation as

Extension of fractional derivative operator
In this section, we recall the definition of following fractional derivatives and give a new extension called Riemann-Liouville k-fractional derivative.

Definition 2.1. The well-known R-L fractional derivative of order µ is defined by
(2.1) In the following, we define Riemann-Liouville k-fractional derivative of order µ as Definition 2.2.
Note that for k = 1, definition 2.2 reduces to the classical R-L fractional derivative operator given in definition 2.1.Now, we are ready to prove some theorems by using the new definition 2.2.
Theorem 2.1.The following formula holds true, Proof.From (2.3), we have Substituting t = uz in (2.6), we get Applying definition (1.9) to the above equation, we get the desired result.
Theorem 2.2.Let ℜ(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| < ρ for some ρ ∈ R + .Then Proof.Using the series expansion of the function f (z) in (2.3) gives As the series is uniformly convergent on any closed disk centered at the origin with its radius smaller then ρ, therefore the series so does on the line segment from 0 to a fixed z for |z| < ρ.Thus it guarantee terms by terms integration as follows which is the required proof.
Proof.By direct calculation, we have Substituting t = zu in the above equation, we get Applying (1.14) and after simplification we get the required proof.
Theorem 2.4.The following result holds true: where To prove(2.9),we use the power series expansion Now, applying Theorem 2.1, we obtain In view of (1.16), we get ) .
Proof.Using the power series for (1 − kz) − α k and applying Theorem 2.5 with η = nk, we can write which is the required proof.[2]) defined as: Proof.Using (2.13), the left-hand side of (2.12) can be written as .
By Theorem 2.2, we have In view of Theorem 2.1, we get the required proof.

.15)
Proof.Applying Theorem 2.1 and followed the same procedure used in Theorem 2.7, we get the desired result.

Concluding remarks
In this paper, we established k-fractional derivative operator.If letting k → 1 then all the results established in this paper will reduce to the results related to the classical Reimann-Liouville fractional derivative operator.