Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 9924265, 11 pages
https://doi.org/10.1155/2021/9924265

Research Article

Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

Mohamed Abdalla,1,2 Muajebah Hidan,¹ Salah Mahmoud Boulaaras,3,4 and Bahri-Belkacem Cherif3,5

¹Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia
²Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt
³Department of Mathematics, College of Sciences and Arts, Qassim University, Arras, Saudi Arabia
⁴Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, Algeria
⁵Preparatory Institute for Engineering Studies in Sfax, Sfax, Tunisia

Correspondence should be addressed to Bahri-Belkacem Cherif; bahi1968@yahoo.com

Received 29 March 2021; Accepted 9 May 2021; Published 25 May 2021

Academic Editor: A. M. Nagy

Copyright © 2021 Mohamed Abdalla et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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., [1–3]). 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, 4–6]). 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 by

which is absolutely and uniformly convergent if , divergent when , and absolutely convergent when , if , where are complex parameters with , and

is 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. [18–20], 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-Riemann–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 by

where . We note that , for , where is the classical Euler’s gamma function and is the k-Pochhammer symbol given in the form

The relation between and gamma function follows easily that

Definition 2. (see [16, 26]). For and , the k-beta function is defined by

where 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 in

where 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 holds

The k-hypergeometric differential equation of second order is defined in [18, 25, 26, 28] by

Particular 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 by

Furthermore, 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) as

where

By using the ratio test and according to the identity , we see that

Thus, 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)–(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)–(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)–(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:

and

where .

Proof. Result (26) is obviously valid in the trivial case when . For , by the power series representation (14) of , we see from (26) that

Replacing the k-Pochhammer symbols by relation (5), we arrive at

Therefore, 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:

and

where .

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. 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 consider

where and is defined in (19).

Similarly, we can write as

where . 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 that

Thus, we get the following differential equation:

Similarly, we can derive the following result of the -extended Kummer hypergeometric function :

In addition, we consider

Replacing by and according to the identity , we have

We 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, 30–34]).

The k-Riemann–Liouville fractional integral using k-gamma function is defined in [20, 21]

In current section, we consider compositions of the k-Riemann–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 at

Changing order of summation and integration and applying relation (7), we obtain

We 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 that

Then, 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.

References

P. Agarwal, R. Agarwal, and M. Ruzhansky, Special Functions and Analysis of Differential Equations, CRC Press, Boca Raton, FL, USA, 1st edition, 2020.

V. Akhmedova and E. Akhmedov, Selected Special Functions for Fundamental Physics, Springer, Berlin, Germany, 2019.

I. Sneddon, Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, Edinburgh, UK, 1956.

S. Jaeyeong and M. Heo, “On MELLIN-Barnes integral representations for GKZ hypergeometric functions,” Kyushu Journal of Mathematics, vol. 74, pp. 109–125, 2020.

M. Abdalla and M. Hidan, “Investigation of the k-analogue of Gauss hypergeometric functions constructed by the Hadamard product,” Symmetry, vol. 13, no. 4, p. 714, 2021.

J. Seaborn, Hypergeometric Functions and Their Applications, Springer, New York, NY, USA, 1991.

H. M. Srivastava, G. Rahman, and K. S. Nisar, “Some extensions of the Pochhammer symbol and the associated hypergeometric functions,” Iranian Journal of Science and Technology, Transactions A: Science, vol. 43, no. 5, pp. 2601–2606, 2019.

H. M. Srivastava, A. Tassaddiq, G. Rahman, K. S. Nisar, and I. Khan, “A new extension of the τ-Gauss hypergeometric function and its associated properties,” Mathematics, vol. 7, no. 10, p. 996, 2019.

R. K. Jana, B. Maheshwari, and A. K. Shukla, “Note on extended hypergeometric function,” Proyecciones (Antofagasta), vol. 38, no. 3, pp. 585–595, 2019.

R. Jana, B. Maheshwari, and A. Shukla, “Some results on the extended hypergeometric function,” Journal of the Indian Mathematical Society, vol. 87, p. 70, 2020.

A. Goswami, S. Jain, P. Agarwal, and S. Araci, “A note on the new extended beta and Gauss hypergeometric functions,” Applied Mathematics & Information Sciences, vol. 12, no. 1, pp. 139–144, 2018.

P. Agarwal, J. Choi, and S. Jain, “Extended hypergeometric functions of two and three variables,” Communications of the Korean Mathematical Society, vol. 30, no. 4, pp. 403–414, 2015.

H. Fuli, A. Bakhet, M. Hidan, and M. Abdalla, “On the extended hypergeometric matrix functions and their applications for the derivatives of the extended Jacobi matrix polynomial,” Mathematical Problems in Engineering, vol. 2020, Article ID 4268361, 8 pages, 2020.

M. Abdalla, “Special matrix functions: characteristics, achievements and future directions,” Linear and Multilinear Algebra, vol. 68, no. 1, pp. 1–28, 2020.

M. Abdalla and A. Bakhet, “Extended Gauss hypergeometric matrix functions,” Iranian Journal of Science and Technology, Transactions A: Science, vol. 42, no. 3, pp. 1465–1470, 2018.

R. Diaz and E. Pariguan, “On hypergeometric functions and k-Pochhammer symbol,” Divulgaciones Mathematics, vol. 15, Article ID 179192, 2007.

P. Agarwal, S. K. Ntouyas, S. Jain, M. Chand, and G. Singh, “Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform,” Alexandria Engineering Journal, vol. 57, no. 3, pp. 1937–1942, 2018.

S. Mubeen, M. Naz, A. Rehman, and G. Rahman, “Solutions of k-hypergeometric differential equations,” Journal of Applied Mathematics, vol. 2014, Article ID 128787, 13 pages, 2014.

S. Mubeen, G. Rahman, A. Rehman, and M. Naz, “Contiguous function relations for k-hypergeometric functions,” ISRN Mathematical Analysis, vol. 2014, Article ID 410801, 6 pages, 2014.

S. Mubeen and G. Habibullah, “An integral representation of some k-hypergeometric functions,” International Mathematical Forum, vol. 7, pp. 203–207, 2012.

G. Rahman, S. Mubeen, and K. S. Nisar, “On generalized k-fractional derivative operator,” AIMS Mathematics, vol. 5, pp. 1936–1945, 2020.

S. Chinra, V. Kamalappan, M. Rakha, and A. Rathie, “On several new contiguous function relations for k-hypergeometric function with two parameters,” Communications of the Korean Mathematical Society, vol. 32, 2017.

D. Korkmaz-Duzgun and E. Erkus-Duman, “Generating functions for k-hypergeometric functions,” International Journal of Applied Physics and Mathematics, vol. 9, pp. 119–126, 2019.

K. Nisar, F. Qi, G. Rahman, S. Mubeen, and M. Arshad, “Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function,” Journal of Inequalities and Applications, vol. 2018, p. 135, 2018.

S. Li and Y. Dong, “k-hypergeometric series solutions to one type of non-homogeneous k-hypergeometric equations,” Symmetry, vol. 11, p. 262, 2019.

Ö. Yilmaz, R. Aktaş, and F. Tasdelen, “On some formulas for the k-analogue of Appell functions and generating relations via k-fractional derivative,” Fractal and Fractional, vol. 4, p. 48, 2020.

M. Hidan, S. Boulaaras, B. Cherif, and M. Abdalla, “Further results on the (p; k) analogue of hypergeometric functions associated with fractional calculus operators,” Mathematical Problems in Engineering, vol. 2021, Article ID 5535962, 9 pages, 2021.

R. Yilmazer and K. Ali, “Discrete fractional solutions to the k-hypergeometric differential equation,” Mathematical Methods in the Applied Sciences, vol. 18, 2020.

M. T. Mohamed, “Certain functions and sets of polynomials and their convergence properties,” Assiut University, Asyut, Egypt, 2003, Ph.D. dissertation.

S. Naz and M. N. Naeem, “On the generalization of k-fractional Hilfer-Katugampola derivative with Cauchy problem,” Turkish Journal of Mathematics, vol. 45, pp. 110–124, 2021.

E. Mittal and S. Joshi, “Note on a k-generallised fractional derivative,” Discrete and Continuous Dynamical Systems Series S, vol. 13, pp. 797–804, 2020.

V. Kiryakova, “Unified approach to fractional calculus images of special functions—a survey,” Mathematics, vol. 8, p. 136, 2020.

M. Zayed, M. Abul-Ez, M. Abdalla, and N. Saad, “On the fractional order Rodrigues formula for the shifted Legendre-type matrix polynomials,” Mathematics, vol. 8, p. 136, 2020.

M. Abdalla, “Fractional operators for the Wright hypergeometric matrix functions,” Advances in Differential Equations, vol. 2020, p. 246, 2020.