#### Abstract

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

#### 1. 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., [1ā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, 4ā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.

Throughout this research, denotes the set of natural numbers, , denotes the set of negative integers, , denotes the set of positive real numbers, and denotes the set of complex numbers.

Traditionally, the hypergeometric function known as Gauss function is defined bywhich is absolutely and uniformly convergent if , divergent when , and absolutely convergent when , if , where are complex parameters with , andis the Pochhammer symbol (or the shifted factorial) and is gamma function. The 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 Euler-type integrals, are associated with properties and applications.

In particular, Diaz and Pariguan [16] introduced the k-analogue of gamma, beta, and hypergeometric functions and proved a number of their properties. Since that period, many different results concerning the k-hypergeometric function and related functions have been considered by many researchers, for instance, Agarwal et al. [17], Mubeen et al. [18ā20], Rahman et al. [21], Chinra et al. [22], Korkmaz-Duzgun and Erkus-Duman [23], Nisar et al. [24], Li and Dong [25], Yilmaz et al. [26], Hidan et al. [27], and Yilmazer and Ali [28].

Motivated by some of these aforesaid studies of the k-hypergeometric functions and related functions, we introduce the -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.

The 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 -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-RiemannāLiouville fractional integral operators of these functions are presented in Section 6. Finally, we point out outlook and observations in Section 7.

#### 2. Preliminaries

In this section, we give some basic definitions and terminologies which are used further in this manuscript.

Definition 1. (see [16, 26]). For , the k-gamma function is defined bywhere . We note that , for , where is the classical Eulerās gamma function and is the k-Pochhammer symbol given in the formThe relation between and gamma function follows easily that

Definition 2. (see [16, 26]). For and , the k-beta function is defined bywhere and .
Clearly, the case in (7) reduces to the known beta function , and the relation between the k-beta function and the original beta function is

Definition 3. (see [16, 26, 28]) Let and and ; then, k-Gauss hypergeometric function is defined inwhere is the k-Pochhammer symbol defined in (5). Obviously, if , equation (9) is reduced to (1).

Proposition 1. (see [16, 26]) For any and , the following identity holdsThe k-hypergeometric differential equation of second order is defined in [18, 25, 26, 28] byParticular choices of the parameters , and 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 such that and ; then, the integral formula of the k-hypergeometric function is given byFurthermore, the k-Kummer (confluent) hypergeometric function is defined in [24] in the form

#### 3. The -Extended Hypergeometric Functions

In this section, we introduce and discuss the -extended Gauss hypergeometric function and -extended Kummer (or confluent) hypergeometric function as follows:where and and , is a positive integer, and is the k-Pochhammer symbol defined in (5).

Remark 1. Some important special cases of and for some particular choice of the parameters and are enumerated below:(1)Putting , we produce the k-analogue of Gauss and Kummer hypergeometric functions which are given in (9) and (13), respectively.(2)Setting , we obtain a -extension of the Gauss and Kummer hypergeometric functions in the following forms, respectively (see Chapter 3 in [29]):(3)Taking and in (11), we produce the standard Gauss hypergeometric function in (1).(4)When and , (12) yields the following special case (see, e.g., [1, 2]).The following theorem shows the convergence property of series (14).

Theorem 2. For all and , the -extended Gauss hypergeometric function given by (14) is an entire function.

Proof. For this prove, we relabel and write (14) aswhereBy using the ratio test and according to the identity , we see thatThus, the power series (14) is convergent for all , under the hypothesis , and . Thus, it yields our desired result.
The following result can be verified in a similar way.

Theorem 3. For all and , the -extended Kummer hypergeometric function given by (15) is an entire function.

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

Remark 2. For in Theorems 2 and 3, we get the convergence property of the k-Gauss hypergeometric function and the k-Kummer hypergeometric function , provided that and (see [16]).

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

#### 4. Integral Representations and Derivative Formulae

##### 4.1. Integral Representations

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

Theorem 4. The following integral representation for in (14) holds true:where .

Proof. Considering the following elementary identity involving the k-Beta function ,in (14), and using relation (17), we get the required integral formula (21).

Theorem 5. The following integral representation for in (14) holds true:where .

Proof. Inserting the k-Pochhammer symbol from (5) in definition (14) by its integral form given by (4) and from relation (15), we thus obtain the desired result (23).
By virtue of the same theorems, we give the following theorem:

Theorem 6. The following integral representations for in (15) hold true:where ,where .

Remark 4. The substitution in (21)ā(25) leads to the integral representations of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).

Remark 5. The substitution in (21)ā(25) leads to the integral formulas of the k-analogue of Gauss and Kummer hypergeometric functions (see [20, 24, 26]).
Also, the special cases of (3.6)ā(3.9) when and are seen to yield the classical integral representations of the Gauss and Kummer hypergeometric functions (see, e.g., [1, 2, 6]).

##### 4.2. Derivative Formulae

Theorem 7. The following derivative formulas hold true:andwhere .

Proof. Result (26) is obviously valid in the trivial case when . For , by the power series representation (14) of , we see from (26) thatReplacing the k-Pochhammer symbols by relation (5), we arrive atTherefore, the general result (26) can now be easily derived by using the principle of mathematical induction on .
A similar procedure yields the desired representation (27).

Theorem 8. The following derivative formulas hold true:andwhere .

Proof. By using series (14) in (30) and differentiating term by term under the sign of summation, we observe thatwhich, in view of series (14), yields the coveted formula (30).
Similarly, we can derive the derivative formula (31).

Remark 6. The special cases of (30) and (31) when are easily seen to reduce to the known derivative formulas of the k-Gauss and Kummer hypergeometric functions (see [18]).

Remark 7. If we take and in the abovementioned theorems, we obtain the corresponding results for the classical hypergeometric functions and (cf. [6]).

#### 5. Contiguous Function Relations and Differential Equations

The k-analogue of theta operator , as given in [18, 19, 25], takes the form . This operator has the particularly pleasant property that , 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 -extended Gauss hypergeometric function and -extended Kummer hypergeometric function .

To realize that, we increase or decrease one or more of the parameters of the -extended Gauss hypergeometric function:

By , then the resultant function is said to be contiguous to . For simplicity, we use the following notations:

Now, we considerwhere and is defined in (19).

Similarly, we can write aswhere . Similarly, for , and .

By the help of differential operator , we get the following relations:

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

Remark 8. Other contiguous function relations for the k-Gauss hypergeometric function may be derived from the relations in (38) to (41) and the same manner, and other results can also be obtained.

Remark 9. We can easily obtain many known results in [19, 22] by setting the parameters in our main findings. Therefore, the obtained results here extend to those results.

Remark 10. It is easy to see that, in (38) to (41), if we take , we get hypergeometric contiguous function relations (see [6]).
Furthermore, the operator , which is used in the derivation of the contiguous function relations, is also used in deriving the differential equations satisfied by and as follows:Using the following identity , we find thatThus, we get the following differential equation:Similarly, we can derive the following result of the -extended Kummer hypergeometric function :In addition, we considerReplacing by and according to the identity , we haveWe thus get the following differential equation:

Remark 11. For in (48), it obviously reduces to the usual differential equation of the k-Gauss hypergeometric function in (11) (see, [18]).
A similar procedure yields differential equation of the -extended Gauss hypergeometric function , by using the operator in (15). We thus obtain differential equation of the -extended Kummer hypergeometric function in the form

Remark 12. The special cases of (48) and (49) when and are seen to yield the classical differential equations of Gauss and Kummer hypergeometric functions (see, for details, [6]).

#### 6. The k-Fractional Integral Operators

Nowadays, computations of images of the k-analogues of special functions under operators of k-fractional calculus have found significant importance and applications by many references (for instance, see, [20, 21, 30ā34]).

The k-RiemannāLiouville fractional integral using k-gamma function is defined in [20, 21]

In current section, we consider compositions of the k-RiemannāLiouville fractional integral operators of the -extended Gauss and Kummer hypergeometric functions.

Theorem 9. Assume that , . Then, for , the following formula holds true:

Proof. By virtue of relations (14) and (50) and putting , we arrive atChanging order of summation and integration and applying relation (7), we obtainWe thus get the required formula (51).

Theorem 10. Assume that , . Then, for , the following formula holds true:

Proof. The proof here would run in parallel with that of Theorem 9. The details are omitted.

Theorem 11. Let , . Then, for , the following formula holds true:where and have the relation

Proof. Assume thatThen, the proof would flow along the lines of that of Theorem 2.4 in [31]. The details are omitted.

Remark 13. The present study in the above theorems is assumed to be extensions of the results in [31].

#### 7. Concluding Remarks

Recently, many studies and extensions of the well-known special functions have been considered by various researchers.

In this paper, we obtained a new extension of the Gauss and Kummer hypergeometric functions, so-called -extended Gauss and Kummer hypergeometric functions. Also, we gave some of their main properties, namely, the convergence properties, integral representations, differential formulas, contiguous function relations, differential equations, and fractional integral operators.

We have spotted that, by setting , the various outcomes presented in this article will reduce to the corresponding outcomes derived earlier in [16, 18, 19, 24, 25, 31]. Furthermore, if we let , then we obtain several interesting new outcomes for the p-extended Gauss and Kummer hypergeometric functions. Finally, we have spotted that if and , then we obtain some known results for the usual Gauss and Kummer hypergeometric functions defined and established in [1, 2, 6]. Additional research and application on this topic is now under preparation and will be presented in forthcoming articles.

#### Data Availability

No data were used to support the study.

#### Conflicts of Interest

This work does not have any conflicts of interest.

#### Acknowledgments

The first-named author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding work through research groups program under Grant no. R.G.P.2/1/42.