A NEW EXTENSION OF EXTENDED CAPUTO FRACTIONAL DERIVATIVE OPERATOR

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.


Extension of hypergeometric functions
In this section, we define further extension of hypergeometric and Appell's hypergeometric functions.
Definition 2.1.The extension of extended hypergeometric function is defined as: Definition 2.2.The extension of Appell's hypergeometric function is defined as: The Integral representations of (2.1) and (2.2) are defined respectively as: and Remark 2.1.If we letting α = 1, then (2.1)-(2.4)reduces to the extended hypergeometric functions 2 F 1 and F 1 and their integral representations (see [6]) respectively.

Extension of fractional derivative operator
Recently, many papers (see [1,8,12,16,21]) introduced various extension and generalization of fractional calculus .In this section, we define a new extension of extended Caputo fractional derivative operator.
We recall the classical Caputo fractional derivative operator which is defined by Definition 3.1.(see [7]) Recently Kiymaz et al. [6] introduced the extended Caputo fractional derivative operator as: The extension of extended Caputo fractional derivative operator [18] defined by Definition 3.3.
In view of [17], we introduce a new extension Caputo fractional derivative operator and is defined as: ) Remark 3.1.Obviously if (i) we letting α = 1, then definition 3.3 reduces to extended Caputo fractional derivative operator defined in 3.2 (see [6]).
(ii) we letting α = 1 and p = 0, then definition 3.3 reduces to extended Caputo fractional derivative operator defined in 3.1 (see [7]).Now, we prove some theorems involving the modified extension of fractional derivative operator.
Proof.From (3.4), we have Substituting t = uz in (3.6), we have By using the definition (1.13) to the above equation, we get which is the required result.Proof.Using the series expansion of the function f (z) in (3.4) gives As the series is uniformly convergent and the integrand is absolutely convergent, therefore interchanging the order of summation and integration gives which is the required proof.
Theorem 3.3.Let m − 1 < (µ) < m and suppose that the function f (z) is analytic on the disk |z| < r for some r ∈ R + and with its power series expansion given by f Proof.By applying Theorems (3.2) and (3.1), we have which is the desired result.
Theorem 3.4.The following result holds true: Proof.Using the power series of (1 − z) −β and applying Theorem 3.1, we have with the aid of (2.1), we get the required result.
Theorem 3.5.The following result holds true: Proof.To prove(3.10),we use the following power series expansion Now, applying Theorem 3.4, we obtain Using Theorem 3.1, we have with the aid of (2.2), we get the required result.

Further results of extended Caputo fractional derivative operator
In this section, we apply the extension of Caputo fractional derivative operator (3.4) to some known functions.Also, we investigate the Mellin transforms of the extension of Caputo fractional derivative operator.
Theorem 4.1.The following result holds true: for all z.
Proof.Using the power series of e z and applying Theorems 3.2 and 3.1, we have Theorem 4.2.The following result holds true: for all |z| < 1.
Proof.Applying Theorem 4.5 with η = n and using the power series extension of (1−z) −α , we can write which is the required proof.

Generating Relations
In this section, we applying Theorems 3.4 and 3.5 and obtain generating relations for the extension of extended hypergeometric functions 2 F α 1,p and F α 1,p .
Proof.By considering the following series identity, we have Thus, the power series expansion yields Multiplying both sides of (5.2) by z η−1 and then applying the operator D η−µ;α z;p on both sides, we have Applying Theorem 3.5 on both sides, we get the desired result.

Concluding remarks
In this paper, we established further extension of Caputo fractional derivative operator and obtained many results related to some known special functions and generating relations via special functions.We conclude that when α = 1 then all the results established in this paper will reduce to the results associated with extended Caputo fractional derivative operator defined by Kiymaz et.al. [6].Similarly, if α = 1 and p = 0 then all the results established in this paper will reduce to the results associated with classical Caputo fractional derivative operator (see [20]).