Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

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


Introduction
Special functions are important tools in solving certain problems arising from many different research areas in mathematical physics, astronomy, chemistry, applied statistics, and engineering (see, e.g., [1][2][3]). Hypergeometric functions are among most important special functions mainly because they have a lot of applications in a variety of research branches such as (for example) quantum mechanics, electromagnetic field theory, probability theory, analytic number theory, and data analysis (see, e.g., [1,2,[4][5][6]). Also, a number of elementary functions and special polynomials are expressed in terms of hypergeometric functions. Accordingly, a number of various extensions of hypergeometric functions have been introduced and investigated. Traditionally, the hypergeometric function known as Gauss function is defined by which is absolutely and uniformly convergent if |v| < 1, divergent when |v| > 1, and absolutely convergent when |v| � 1, if Re(δ 3 − δ 1 − δ 2 ) > 0, where δ 1 , δ 2 , and δ 3 are complex parameters with δ 3 ∈ C\Z − 0 , and is the Pochhammer symbol (or the shifted factorial) and Γ(.) is gamma function. e function in (1) satisfies the following differential equation: Nowadays, numerous investigations, for example, in recent works of Srivastava et al. [7,8], Jana et al. [9,10], Goswami et al. [11,12], Fuli et al. [13], and Abdalla and Bakhet [14,15] to introduce extensions and generalizations of the hypergeometric functions, defined by Eulertype integrals, are associated with properties and applications.
Motivated by some of these aforesaid studies of the k-hypergeometric functions and related functions, we introduce the (p, k)-extended Gauss and Kummer hypergeometric functions and their properties. Relevant connections of some of the discussed results here with those presented in earlier references are outlined. e manuscript is organized as follows. In Section 2, we list some basic definitions and terminologies that are needed in the paper. In Section 3, we introduce the (p, k)-extended Gauss and Kummer (or confluent) hypergeometric functions and discuss their regions of convergence. In Section 4, we obtain integral and differentiation formulas of the (p, k)extended Gauss and Kummer hypergeometric functions. In addition, contiguous function relations and differential equations connecting these functions are established in Section 5. Compositions of the k-Riemann-Liouville fractional integral operators of these functions are presented in Section 6. Finally, we point out outlook and observations in Section 7.

Preliminaries
In this section, we give some basic definitions and terminologies which are used further in this manuscript.
Definition 1 (see [16,26]). For k ∈ R + , the k-gamma function Γ k (u) is defined by where u ∈ C\kZ − . We note that Γ k (u) ⟶ Γ(u), for k ⟶ 1, where Γ(u) is the classical Euler's gamma function and (u) m,k is the k-Pochhammer symbol given in the form e relation between Γ k (u) and gamma function Γ(u) follows easily that Definition 2 (see [16,26]). For u, v ∈ C and k ∈ R + , the where Re(u) > 0 and Re(v) > 0. Clearly, the case k � 1 in (7) reduces to the known beta function B(u, v), and the relation between the k-beta function B k (u, v) and the original beta function B(u, v) is Definition 3 (see [16,26,28]) Let k ∈ R + and s 1 , s 2 , η ∈ C and s 3 ∈ C\Z − 0 ; then, k-Gauss hypergeometric function is defined in where (s 1 ) m,k is the k-Pochhammer symbol defined in (5). Obviously, if k � 1, equation (9) is reduced to (1).
Proposition 1 (see [16,26]) For any δ ∈ C and k ∈ R + , the following identity holds e k-hypergeometric differential equation of second order is defined in [18,25,26,28] by Particular choices of the parameters s 1 , s 2 , s 3 , and k in the linearly independent solutions of the differential equation (11) yield more than 24 special cases. Also, the k-hypergeometric function can be given an integral representation in the following result [20,26]: Theorem 1. Assume that η, s 1 , s 2 , s 3 ∈ C such that Re(s 3 ) > Re(s 2 ) > 0 and k ∈ R + ; then, the integral formula of the k-hypergeometric function is given by 2 Furthermore, the k-Kummer (confluent) hypergeometric function 1 5 k 1 is defined in [24] in the form

The (p, k)-Extended Hypergeometric Functions
In this section, we introduce and discuss the (p, k)-extended Gauss hypergeometric function W(p, k; ξ) and (p, k)-extended Kummer (or confluent) hypergeometric function Y(p, k; ξ) as follows: where k ∈ R + and ζ 1 , ζ 2 , ξ ∈ C and ζ 3 ∈ C\Z − 0 , p is a positive integer, and (ζ) m,k is the k-Pochhammer symbol defined in (5). Remark 1. Some important special cases of W(p, k; ξ) and Y(p, k; ξ) for some particular choice of the parameters p and k are enumerated below: (1) Putting p � 1, we produce the k-analogue of Gauss and Kummer hypergeometric functions which are given in (9) and (13), respectively.

Theorem 2.
For all k ∈ R + and p > 1, the (p, k)-extended Gauss hypergeometric function W(p, k; ξ) given by (14) is an entire function.
Proof. For this prove, we relabel and write (14) as where By using the ratio test and according to the identity (ζ) m+1,k � (ζ + mk)(ζ) m,k , we see that Mathematical Problems in Engineering 3 us, the power series (14) is convergent for all |ξ| < ∞, under the hypothesis p > 1, k ∈ R + , and ζ 3 ∈ C\Z − 0 . us, it yields our desired result. e following result can be verified in a similar way. □ Theorem 3. For all k ∈ R + and p > 1, the (p, k)-extended Kummer hypergeometric function Y(p, k; ξ) given by (15) is an entire function.

Corollary 1.
For all p > 1, the power series (16) and (17) are an entire function.

Remark 3.
For p � 1 in Corollary 1, we obtain the convergence property of the usual Gauss and Kummer hypergeometric series (see [1,2]).

Integral Representations.
In the following, we establish the following theorems in terms of the k-integral representations of the (p, k)-extended Gauss and Kummer hypergeometric functions.

Derivative Formulae
e following derivative formulas hold true: and where k ∈ R + , p > 1, and n ∈ N 0 .
Proof. Result (26) Replacing the k-Pochhammer symbols (ζ 1 + k) m,k by relation (5), we arrive at erefore, the general result (26) can now be easily derived by using the principle of mathematical induction on n ∈ N 0 .
e following derivative formulas hold true: and

Mathematical Problems in Engineering
where k ∈ R + , p > 1, and n ∈ N 0 .
Proof. By using series (14) in (30) and differentiating term by term under the sign of summation, we observe that which, in view of series (14), yields the coveted formula (30).
Similarly, we can derive the derivative formula (31). □ Remark 6. e special cases of (30) and (31) when p � 1 are easily seen to reduce to the known derivative formulas of the k-Gauss and Kummer hypergeometric functions (see [18]).

Contiguous Function Relations and
Differential Equations e k-analogue of theta operator kΘ, as given in [18,19,25], takes the form kΘ � kξ(d/dξ). is operator has the particularly pleasant property that kΘξ m � kmξ m , which makes it handy to be used on power series. In this section, relying on Definition 1, we present some results concerning contiguous function relations and differential equations for the (p, k)-extended Gauss hypergeometric function 2 F To realize that, we increase or decrease one or more of the parameters of the (p, k)-extended Gauss hypergeometric function: ; ξ , k ∈ R + , p > 1.

(37)
From the above relations, we can easily obtain the following results: