ON EXTENDED CAPUTO FRACTIONAL DERIVATIVE OPERATOR

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

In the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as ) dt, (1.4) dt, (1.5) ) .
The extended Appell's function is defined by (see [10]) where p ≥ 0 and its integral representation by It is clear that when p = 0, then the equations (1.2)-(1.7)reduce to the well known hypergeometric, confluent hypergeometric and Appell's series and their integral representation respectively (see [18]).Parmar et al. [15] 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 (1.8) reduces to (1.1) by using the fact that K1 In the same paper, they [15] defined the following extended hypergeometric and confluent hypergeometric functions and their integral representation respectively as: It is clear that, when v = 0 then the equations (1.9)-(1.12)reduce to the extended hypergeometric, confluent hypergeometric functions and their integral representations defined in (1.2)-(1.5)respectively by using the fact that K 1 2 ( z 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: ) dt, (1.14) where Obviously, when v = 0 in (1.13) and (1.14) then we get the extended Appell function and its integral representation (see e. g., (1.6) and (1.7)) by using the fact that K1 Very recently Rahman et al. [17] introduced the extension of extended fractional derivative operator of Riemann-Liouville as: where ℜ(µ) > 0, ℜ(p) > 0 and v ≥ 0. It is clear that, if v = 0, then definition 1.15 reduces to extended fractional derivative defined in [10] by using the fact that

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: )

Extension of fractional derivative operator
Recently, the application and importance of fractional calculus have been paid more attentions.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 [1,8,12,16,20]).In this section, we define further 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: ) In view of [17], we introduce further extension of extended Caputo fractional derivative operator.
Definition 3.3.The extension of extended Caputo fractional derivative operator is defined as; Remark 3.1.Obviously if v = 0, then definition 3.3 reduces to extended Caputo fractional derivative defined in 3.2 (see [6]) by using the fact that K 1 2 ( z Now, we prove some theorems involving the modified extension of fractional derivative operator.
Proof.From (3.3), we have Substituting t = uz in (3.5), we have by using the definition (1.8) to the above equation, we get which is the required result.
Theorem 3.2.Let m − 1 < ℜ(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.Using the series expansion of the function f (z) in (3.3) 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 < ℜ(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 Theorem (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(3.9),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 (3.3) 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.