Further Results on the (p, k)−Analogue of Hypergeometric Functions Associated with Fractional Calculus Operators

Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Buraydah, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, 31000 Oran, Algeria Preparatory Institute for Engineering Studies in Sfax, Sfax, Tunisia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt

In particular, Diaz and Pariguan [8] introduced interesting generalizations of the gamma, beta, Pochhammer, and hypergeometric functions as follows.
e k-beta function B k (z, w) is defined by which is an entire function for p > 1, where k ∈ R + and θ 1 , θ 2 , ξ ∈ C and θ 3 ∈ C\Z − 0 , and (x) s,k is the k-Pochhammer symbol defined in (2). ey studied several its properties such as the k-Euler and Laplace-type integral representations, differentiation type formulae, contiguous function relations, and differential equations (cf. [18] (6) and (7). Furthermore, when k ⟶ 1, we obtained the p-extend hypergeometric function in the following form (see Chapter 3 in [19]): which is an entire function for p > 1. Motivated by some of these aforesaid studies of the (p, k)− extended hypergeometric function 2 H (p,k) 1 defined by (9), in this manuscript, we investigate certain integral representations, a derivative formula, k-Beta transform, Laplace and inverse Laplace transforms, and fractional calculus operators of the (p, k)-extended hypergeometric function which may be useful for carrying out further research studies to make more other developments and extensions of this field. In addition, some interesting special cases of our main results are also indicated.

Integral Representations and Derivative Formula
Starting, we establish the following theorems in terms of the k-integral representations of the (p, k)-extended hypergeometric functions as follows.
Proof. Inserting series (9) in the LHS of (11) and by using relation (2), we obtain Letting u � ηv and according to the k-beta function (4), we find that e above equation gives the RHS of (11). We thus obtain result (11) in eorem 1.

Theorem 2.
e following integral representation for 2 H (p,k) 1 in (9) holds true: Proof. For convenience, let the left-hand side of (14) be denoted by T. Applying the series expression of (9) to T, we observe that Setting Making an appeal to the following duplication k-gamma formula (cf. [8]), Mathematical Problems in Engineering 3 In the above series and after a simplification, we obtain We thus arrive at the desired result (14) Proof. Taking left-hand side of equation (19) by S and v � (u/(u + 1)), we have Changing the order of integration and summation in (20) and by using relation (4), we obtain We thus get the required integral formula (19). Similarly, we can easily obtain the following result without proof.
Remark 2. At p � 1 in eorems 1-4, we obtain integral formulae of the k-analogue of hypergeometric functions defined in (6).

Remark 3.
e substitution k � 1 in eorems 1-4 leads to the integral representations of the p− extended hypergeometric functions defined in (10).
Remark 4. If we take p � 1 and k � 1 in the abovementioned theorems, we obtain the corresponding results for the classical hypergeometric functions (see, e.g., [20]).

Theorem 5.
e following derivative formula holds true: Proof. Result (23)  , we see from (23) that Replacing the k-pochhammer symbol (θ 3 + k) n,k in relation (2) by its k-gamma function and using the k-gamma function property given in [8] erefore, the general result (23) can now be easily derived by using the principle of mathematical induction on m ∈ N 0 . □ Remark 5. p � 1 in (23) leads naturally to the differentiation formula for 2 F k 1 in (6).

Integral Transforms
In this section, we prove three theorems, which exhibit the connection between the integral transforms such as k-Beta transform and Laplace and inverse Laplace transforms for 2 H (p,k) 1 given in (8). in (8) is given in the following form:

Mathematical Problems in Engineering
where the k-Beta transform of Φ(ξ) is defined as Proof. By invoking definition (27) and applying (9) to the k-Beta transform of (26), we obtain 1 k Changing the order of integration and summation, we have 1 k which, upon using (4) and (9), yields our desired result (26) in eorem 6. □ Theorem 7. e following Laplace transform and inverse Laplace transform formulae hold, respectively: where the Laplace transform and is defined as (see Chapter 3 in [21]): For simplicity, we can write the inverse Laplace transform as provided that both sides above results exist.
Proof. To prove (30), we use the power series expansion (9), and applying definition (32), we observe that By change the order of integration and summation with setting wz � (τ k /k) and then applying the k-gamma function (1) to the last integral, we obtain which leads to the desired result (30). Now, in order to demonstrate (31), making use of (8) in the left-hand side of (31), we find that After a simplification, we get the required result in (31).
Here, we consider the k-fractional differentiation of the (p, k)-extend hypergeometric functions using define the extended Riemann-Liouville k-fractional derivative with the parameters δ, μ, ] ∈ C.
We need the following lemma.
Lemma 1 (see [27]). e following formula holds true: Mathematical Problems in Engineering