Fractional Calculus of the Extended Hypergeometric Function

Here, our aim is to demonstrate some formulae of generalization of the extended hypergeometric function by applying fractional derivative operators. Furthermore, by applying certain integral transforms on the resulting formulas and develop a new futher generalized form of the fractional kinetic equation involving the generalized Gauss hypergeometric function. Also, we obtain generating functions for generalization of extended hypergeometric function..

Lemma 3. Let λ , ρ ∈ C. Then, we have the following relations and From its birth to its today's wide use in a great number of scientific fields fractional calculus has come a long way. Despite the fact that its nearly as old as classical calculus itself, it flourished mainly over the last decades because of its good applicability on models describing complex real life problems (see. [5,6,28,41]).
Here, by choosing a known generalization of the extended Gauss hypergeometric function in (7) we aim to establish certain formulas and representations for this extended Gauss hypergeometric function such as fractional derivative operators, integral transforms, fractional kinetic equations and generating functions. Also, we give some generating functions for extended Appell hypergemetric functions (9) and (10).
Definition 1. Let f (z) := ∑ ∞ n=0 a n z n and g(z) := ∑ ∞ n=0 b n z n be two power series whose radii of convergence are given by R f and R g , respectively. Then their Hadamard product is power series defined by ( f * g) := ∞ ∑ n=0 a n b n z n (27) whose radius of convergence R satifies R f .R g ≤ R.
Especially, if one of the power series defines an entire function and the radius of convergence of the grater than zero, then the Hadamard product to seperate a newly-emerged function into two known functions. For example, r F s+m (a 1 ; p, q; κ, µ), a 2 · · · a r ; b 1 , b 2 , · · · , b s+m ; z (|z| < ∞) The main results are obtained in the following theorems. Proof.
Let's denote the left-hand side of the equation (29) by L. Using the definition of the generalized hypergeometric function (7) and arranging order of integration and summation, which is applicable under the conditions Theorem 1, we get taking advantage of the (21) in the above equality (30), we have after simplfying the equation (31), we obtain further comment in the view of (7), we obtain Finally, we have the desired result (29) in consinderation of the equation (28).
Proof. We can obtain the proof of (34) given above similar to Theorem 1.
Applying σ = 0 in the equations (29) and (34) yields some results asserted by the following corollaries.
3 Integral Transforms of (7) In this section, we prensent some integral transforms for example, P δ transform, Laplace transform, Sumudu transform, Hankel transform and Laguerre transform for the generalization of the extended hypergeometric function (7).

P δ and Related Integral Transforms
The P δ transform of f (t) is defined as [14,24] on condition that the convenient existence condition given by Lemma 4 below are satisfied.

Lemma 10.
Let the function f(t) be integrable over any finite interval (a, b) (0 < a < t < b). Suppose also that there exists a real number c such that each of the following assertions holds true: (i) For any arbitrary b > 0,´ι b e −ct f (t)dt tends to a finite limit as ι → ∞; (ii) For any arbitrary a > 0,´a ε | f (t)|dt tends to a finite limit as ε → 0 + . Then the P δ -transform exists whenever where Proof. Let's denote the left-hand side given in equation (40) by P and using the definition of the P δ -transform (39), we get; taking advantage of the equation (29) and arranging order of integration and summation, which is applicable under conditions Theorem 3, we have after simplfying the equation (42), we obtain further comment in the view of (7), we have ).
Finally, we get the required result (40) in consinderation of the equation (28).
Proof. The proof of the Theorem 4 is parallel to the proof of Theorem 3 Upon letting δ → 1+ in the equation (39) is immediately reduced to the classic Laplace transform.
The Laplace transform of f (z) is defined as [14,26,35]: The folowing theorem is a limit case of Theorem 3 and Theorem 4 when δ → 1+ Proof. Let's denote the left-hand side of the equation (47) by L. Using the definition of the Laplace transform in the above equation, we have taking advantage of the equation (29) and arranging order of integration and summation, which is applicable under conditions Theorem 5, we have after simplfying the equation (49), we obtain further comment in the view of (7), we have Finally, we get the required result (47) in consinderation of the equation (28).
Proof. The proof of the Theorem 6 is similar to the proof of Theorem 5.

Now, taking s = 1 in the equation (46) is related to the Sumudu transform.
The Sumudu transform of f (z) is given as follows [39]: The following corollaries is the special case of Theorem 5 and Theorem 6 when s = 1 (54)
Proof. Let's denote the left-hand side of the equation (57) by H. Using the definition of the Whittaker transform in the above equation, we have Now, applying well-knowns formula for power function including Bessel function [14,24], after simplfying equation (59) and using the definition of (7), we obtain Finally, we get the required result (57) in consinderation of the equation (28).
Proof. The proof of the Theorem 8 is similar to the proof of Theorem 7.
Proof. Let's denote the left-hand side of the equation (63) by M. Using the definition of the Laguerre transform in the above equation, we have Now, applying well-knowns integral formula for power function including Laguerre polynomial [14,24], after simplfying equation (65) and using the definition of (7), we obtain Finally, we get the required result (63) in consinderation of the equation (28).
Proof. The proof of the Theorem 10 is similar to the proof of Theorem 9.
4 Generating functions of (7), (9) and (10) In this section, we will present certain generating functions involving new generalization of extended Gauss hypergeometric function and extended Appell hypergeometric functions.
Theorem 21. The following generating function for (7) holds true: Proof. Let's obtain the left hand side of the equation (68) by S. Then, by applting the series expression from (7) into S, we have that which, upon arranging the order of summation and after some changing, gives Finally, applying the following generalized binomial expression [34,35]: for calculating the inner sum in (70), we obtain the required result (68).

Theorem 22.
The following generating function for (7) holds true: (α) n n! F κ,µ p,q (a, α + n; c; z)ω n = (1 − ω) −α F κ,µ p,q (a, α; c; Proof. The proof of the Theorem 12 is same as the proof of Theorem 1. Theorem 23. The following generating function for (9) holds true: Proof. Let N be the left side of (73), using the (9) and interchanging the order of summations, we have that Applying the equation (71) in the equation (74), we can be easily seen to lead to right-hand side of (73).
Theorem 24. The following generating function for (10) holds true: Proof. The proof of the Theorem 14 is same as the proof of Theorem 13.

Fractional Differential Equations
The importance of the fractional differential equations in the field of applied sciences gained more attention not in mathematics but also in physics, dynamical systems, control systems and engineering, to create the mathematical model of physical phenomena. Specially, the kinetic equations describe the contiunity of motion of substance. The extension and generalisation of fractional kinetic equations involving many fractional operators were found in [2,3,9,11,12,15,[31][32][33].
The fractional differential equation between rate of change of the reaction, the destruction rate and the production rate was established by Haubold and Mathai [15] given as follows: where N = N(t) the rate of the reaction, d = d(N) the rate of destruction, p = p(N) the rate of production and N t denotes the function defined by N t (t ) = N(t − t ),t > 0. The special case of equation (76) for spatial fluctuations and inhomogeneities in N(t) the quantities are neglected, that is the equation with the initial condition that N i (t = 0) = N 0 is the number of density of the species i at time t = 0 and c i > 0. If we shift the index i and integrate the standard kinetic equation (77), we have where 0 D −1 t is the special case of the Riemann-Liouville integral operator 0 D −ν t given as The fractional generalisation of the standart kinetic equation (78) is given by Haubold and Mathai as follows [31,32]: and obtained the solution of (77) as follows: Furthermore, Saxena and Kalla [33] considered the following fractional kinetic equation: where N(t) denotes the number density of a given species at time t, N 0 = N(0) is the number of density of that species at time t = 0, c is a constant and f ∈ L(0, ∞). By applying the Laplace transform (46) to the equation (82),

Solution of the generalised fractional kinetic equations
In this section, we will present the solution of the generalised fractional kinetic equations which by considering generalised Gauss hypergeometric function (7).
So, we can be yield the required result (84).

Conclusions
We may also give point to that results obtained in this work are of general character and can appropriate to give farther interesting and potentially practical formulas involving integral transform, fractional calculus and generating functions. Also we give a new fractional generalization of the standard kinetic equation and obtained solution for the same. From the close relationship of family of extended generalized Gauss hypergeometric functions with many special functions, we can easily construct various known and new fractional kinetic equations.