On some formulas for the $k$-analogue of Appell functions and generating relations via $k$-fractional derivative

Our present investigation is mainly based on the $k$-hypergeometric functions which are constructed by making use of the Pochhammer $k$-symbol \cite{Diaz} which are one of the vital generalization of hypergeometric functions. We introduce $k$-analogues of $F_{2}\ $and $F_{3}$ Appell functions denoted by the symbols $F_{2,k}\ $and $F_{3,k}\ $respectively, just like Mubeen et al. did for $F_{1}$ in 2015 \cite{Mubeen6}. Meanwhile, we prove some main properties namely integral representations, transformation formulas and some reduction formulas which help us to have relations between not only $k$-Appell functions but also $k$-hypergeometric functions. Finally, employing the theory of Riemann Liouville $k$-fractional derivative \cite{Rahman} and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for $k$-analogue of hypergeometric functions and Appell functions.


Introduction
Special functions, with its diverse sub-branches, are very wide field of study and are used not only in various fields of mathematics but also in the solutions of important problems in many disciplines of science such as physics, chemistry and biology. This subject is powerful to make sense of uncertain questions especially in physical problems so it encourages many people for notable improvements on this matter. As in other sciences, remarkable problems are still discussed in many disciplines and more general results are tried to be obtained.
Many elementary functions can be expressed in terms of hypergeometric functions. Moreover; non-elementary functions that occur in physics and mathematics have a representation with hypergeometric series. Therefore, generalizing hypergeometric functions bring with different generalizations in other disciplines. These generalizations can be made by increasing the number of parameters in the hypergeometric function or by increasing the number of variables. Appell, based on the idea that the number of variables can be increased, has defined Appell hypergeometric functions obtained by multiplying two hy-pergeometric functions. These are the four elemanter functions defined in [4,5] F 1 (α, β, β ′ ; γ; x, y) = ∞ m,n=0 where |x| < 1, |y| < 1, |x| + |y| < 1, |x| < 1, |y| < 1, |x| + |y| < 1, respectively.
Another generalization of hypergeometric functions is the hypergeometric k -function, defined by the Pochhammer k-symbol studied by Diaz et al. [1]. This paper includes the k-analogue of the Pochhammer symbol and hypergeometric function, as well as the k-generalization of gamma, beta, and zeta functions with their integral representations and some identities provided by classical ones. It should be noted that, taking k = 1 in these generalizations, the k-extensions of the functions reduce to the classical ones.
Let k ∈ R + and n ∈ N + . Hypergeometric k-function is defined in [1] as where α, β, γ, x ∈ C and γ neither zero nor a negative integer and (λ) n,k is the Pochhammer k-symbol defined in [1] as Based on this generalization, Kokologiannaki [6] obtained different inequalities and properties for the generalizations of Gamma, Beta and Zeta functions.
Some limits with the help of asymptotic properties of k-gamma and k-beta functions were discussed by Krasniqi [7]. Mubeen et al. [8] established integral representations of the k-confluent hypergeometric function and k-hypergeometric function and in another paper [9], proved the k-analogue of the Kummer's first formulation using these integral representations. In [10], some families of multilinear and multilateral generating functions for the k-analogue of the hypergeometric functions were obtained. Studies on this subject are not limited to these papers, for detailed [11,12,13,14].
In [15], Mubeen adapted the k-generalization to the Riemann Liouville fractional integral by using k-gamma function. In [16], Our present investigation is motivated by the fact that generalizations of hypergeometric functions have considerable importance due to their applications in many disciplines from different perspectives. Therefore, our study is generally based on the k-extension of hypergeometric functions. The structure of the paper is organized as follows: In section 2, we briefly give some definitions and preliminary results which are essential in the following sections as noted in [1,15,2]. In section 3, following [1,2] and using the same notion, we are concerned with the k-generalizations of F 2 and F 3 Appell hypergeometric functions. Moreover, we prove some main properties such as integral representations, transformation formulas and some reduction formulas which enables us to have relations for khypergeometric functions and k-Appell functions. In the last part of the paper, applying the theory of Riemann Liouville k-fractional derivative [3] and using the relations which we consider previous sections, we gain linear and bilinear generating relations for k-analogue of hypergeometric functions and k-Appell functions.

Some Definitions and Preliminary Results
For the sake of completeness, it will be better to examine the preliminary section in three subsections by reason of the number of theorems and definitions. In these subsections, we will present some definitions, properties and results which we need in our investigation in further sections. We begin by introducing k-gamma, k-beta and k-analogue of hypergeometric function and we continue definition of k-generalization of F 1 which is the first Appell function.
We conclude this section with recalling Riemann Liouville fractional derivative, k-generalization of this fractional derivative and some important theorems which will be required in our studies.
Through this paper, we denote by C, R, R + , N and N + the sets of complex numbers, real numbers, real and positive numbers and positive integers with zero and positive integers, respectively.

k-Generalizations of Gamma, Beta and Hypergeometric Functions
In this subsection, we will present the definitions of k-gamma and k-beta functions are presented and some elemental relations provided by these functions are introduced by Diaz et al. [1] and Mubeen et al. [12]. Furthermore, we continue the definition of k-hypergeometric function and we present integral representation and some formulas satisfied from this generalization [8,9].
Definition 1. For x ∈ C and k ∈ R + , the integral representation of k-gamma function Γ k is defined by where ℜ (x) > 0 [1,12].
Definition 2. For x, y ∈ C and k ∈ R + , the k-beta function B k is defined by where ℜ (x) > 0 and ℜ (y) > 0 [1]. Proposition 1. Let k ∈ R + , a ∈ R, n ∈ N + . The k-gamma function Γ k and the k-beta function B k satisfy the following properties [1,12], Definition 3. Let x ∈ C, k ∈ R + and n ∈ N + . Then the Pochhammer k-symbol is defined in [1,12] by In particular we denote (x) 0,k := 1.
Assume that x ∈ C, k ∈ R + and ℜ (γ) > ℜ (β) > 0, then the integral representation of the k-hypergeometric function is defined in [8] as For the following theorem, 2 Theorem 2. [9] Assume that x ∈ C, k ∈ R + and Re (γ − β) > 0, then For the special case α = −n, Here, we remind the definition of k-analogue of F 1 which is the first Appell function and some identities which are satisfied by it [2].
integral representation of the k-hypergeometric function is as follows

The Riemann Liouville k-Fractional Derivative Operator
Fractional calculus and its applications have been intensively investigated for a long time by many researches in numerous disciplines and its attention has grown tremendously. By making use of the concept of the fractional derivatives and integrals, various extensions of them has been introduced and authors have gained different perspectives in many areas such as engineering, physics, economics, biology, statistics [17,18]. One of the generalization of fractional derivatives is Riemann Liouville k-fractional derivative operator studied in [3,16,19].
Here, we remind the definition of Riemann Liouville fractional derivative and its k-generalization and also some theorems which will be used in further section, are shown.

Definition 5. [5]
The well known Riemann Liouville fractional derivative of order µ is described, for a function f, as follows where ℜ (µ) < 0.
In particular, for the case where ℜ (µ) < 0 and k ∈ R + .
In particular, for the case m − 1 < ℜ (µ) < m where m = 1, 2, ..., (29) is written by Let Re (µ) > 0 and suppose that the function f (z) is analytic at the origin with its Maclaurin expansion has the power series expansion Then the following result holds true

k-Generalizations of the Appell Functions and Some Transformation Formulas
In 2015, k-generalization of F 1 Appell function was introduced and contiguous function relations and integral representation of this function were shown by using the fundamental relations of the Pochhammer k-symbol [2]. The kanalogue of the F 1 was defined but other Appell k-functions such as F 2 , F 3 and F 4 have not yet been explored. We now turn our attention the definition of F 2 and F 3 and provide the integral representation of them. Also we derive some linear transformations of Appell functions and give some reduction formulas involving the 2 F 1,k hypergeometric function.
Also, the first Appell k-function F 1,k defined by (25) is expressed in terms of 2 F 1,k as follows As a first theorem, we consider the integral representation F 2,k and F 3,k . We note that the integral representation of F 1,k can be found [2].
Theorem 8. Let k ∈ R + . Integral representations of F 2,k and F 3,k have the forms of Proof. From the definition of Pochhammer k-symbol, we can write We insert these formulas with the integral representation of B k into the definition of F 2,k given by (36), we find that which completes the proof.
Formula (41) can be proved in a similar way, hence the details are omitted.
Theorem 9. For k ∈ R + , F 1,k has the following relation Proof. In [2], the integral representation of F 1,k is given by Performing change of variables t = 1 − t 1 in above integral, we can write Thus we get the desired result.
Theorem 10. For k ∈ R + , we have Proof. By a change of variables using t = t1 1−kx+kt1x in the integral representa-tion of F 1,k , we have that In the above integral we note that, Using similar argument with t = t1 1−ky−kt1y , one can easily obtain Theorem 11. Let k ∈ R + then F 1,k has the following relations, and F 1,k (α, β, β ′ ; γ; x, y) Proof. Using t = t1 1−kx+kxt1 and t 1 = 1 − t 2 in integral representation of F 1,k , we obtain Using the same method as above, we can reach (47) easily.
We continue with some reduction formulas for Appell functions F 1,k and F 2,k in terms of the 2 F 1,k generalized hypergeometric function.
In the next lemma, we will prove Euler transformation for 2 F 1,k hypergeometric function which will be used in the next theorem.
Proof. From the definition of 2 F 1,k , one gets Using the identity (m − n)! = (−1) n m! (−m) n in (56) , we thus find that Making use of (24) in (57), we get the desired result.
Theorem 14. Let k ∈ R + . Then we have Proof. Using the definition of F 1,k defined by (39) and making use of (55), we can write Thus we finish the proof.

Generating Relations Involving the Generalized Appell Functions
In this section, employing the theory of Riemann Liouville k-fractional derivative [3] and making use of the relations which we consider previous sections, we establish linear and bilinear generating relations for k-analogue of hypergeometric functions and k-Appell functions.

Theorem 15. We have the generating relation
Proof. To prove the result, consider the elementary identities given by

From the series expansion using the definition of Pochhammer
we can write From (60) and (61), we have the equality where |t| < |1 − kx| . Multiplying both sides of (62) by x α k −1 and then applying k D α−β x to the both sides of (62), we can reach Since ℜ(α) > 0 ve |t| < |1 − kx|, it is possible to change the order of the summation and differentiation, we get Finally using relation (34) in (63), it follows Hence, we get the desired result.

Theorem 16. We have the generating relation
Proof. Consider the identity Under the assumption |kt| < |1 − kx| −1 ,we can rewrite (65) − ρ k and taking the D α−β x on both sides of (66), we For ℜ(α) > 0 , interchanging the order of the summation and the operator Assuming |x| < 1 k and kxt 1−kt < 1 k and using (34) and (35), Proof. We use the result of the previous theorem. Setting λ = β − ρ in (64), we If we use reduction formula for F 1,k given by (52), we obtain easily the desired result as follows, Multiplying both sides of (70) with t (β) n,k In view of (31) and (35) on the right and left side of (71), respectively, we can Theorem 18. We have the generating relation Proof. Putting (1 − ky) t instead of t in (59), we can obtain Multiplying with y γ k −1 , employing k D γ−δ y both sides of (73) and the under the assumption ℜ (γ) > 0 interchanging differentiation and summation, we can write Make use of the formula (34), we can easily simplify left side of the (74) as follows, For the right side of the (74), using the definition of 2 F 1,k and the formula (31), one obtain where |x| < 1 k , |y| < 1 k , Combining the relations (75) and (76), we get desired result.
As a special case of (72), we give the following theorem as follows.