Abstract
The present paper establishes several new integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables. The main results are obtained by using the properties of Gamma and beta functions. The novel integral representations are carried out through ten hypergeometric functions of three variables. Therefore, all derived integrals are generalization representation of the Euler type for the classical Gauss hypergeometric function of one and two variables. In addition, several numerical examples were given to describe some of the obtained results.
1. Introduction
Special functions are particular mathematical functions with more or less established names and notations due to their significance in mathematical analysis. The hypergeometric function is one of the essential special functions that have had more attention in recent years, including many other special functions as specific or limiting cases. The hypergeometric functions and their generalized extensions have been introduced and investigated extensively due mainly to their applications in diverse areas in mathematics, applied mathematics, physics, and engineering. Srivastava and Karlsson[1] introduced and analyzed the extension of the generalized hypergeometric function utilizing the extended Pochhammer symbol. Another extension of the generalized hypergeometric functions is defined in reference [2]. Ref. [3] (Ali et al. introduced the multiindex Gauss hypergeometric functions using the multiindex Beta function. Saboor et al. [4] characterized a new extension of Srivastava’s triple hypergeometric functions; besides, they presented their specific properties, derivative formulas, integral representations, and some new recurrence relations. Recently, numerous researchers have studied several extensions and generalizations of many special functions (see, e.g., [5–11]).
It is known that there are 205 hypergeometric functions of three variables of second order, of which regions of convergence have been given in the literature. Hypergeometric functions of three variables can be expressed either by integrals of the Euler type or by integrals of the Laplace type [12]. The list of hypergeometric functions of three variables is too extensive (see [13], Chapter 3), and it is impossible to give a complete list of integral representations here. Integral representations of the hypergeometric functions with three variables are helpful for the analytical continuation [14]. In addition, an analytic continuation of the Horn hypergeometric function with an arbitrary number of variables is given in [15]. Therefore, the integral representations are mainly in the theory of transformation, as well as the integration of hypergeometric systems of partial differential equations (PDEs) (see [5]). In the attending work, the authors aim to obtain some new integral representations for hypergeometric functions of three variables by applying either the first-kind Euler integral or the integral of the second kind.
2. Preliminaries
In this section, the definition and some helpful relations will concern. For our purpose, we begin by recalling some Gauss hypergeometric functions [13] of three variables as follows:
We note that is the well-known Pochhammer’s symbol, and it was handed by
The Gauss hypergeometric function is defined as (see Ref. [12])
The two-variables hypergeometric functions and are, respectively, defined by (see [16, 17])and
Horn’s functions of two variables (see Ref. [18]), namely, , and , are given as
The useful Lauricella functions of three variables (see [13, 19]), namely, , and , are characterized by
Lastly, the three-variable Exton’s functions (see Ref. [20]), namely, , and , are used here and expressed as follows:
Now, we turn to the main results of the work.
3. Integral Representations of Euler Type
In the current section, we present certain Euler-type integral representations of the Gaussian triple hypergeometric functions defined in (1) to (10), whose kernels contain Gauss hypergeometric function , Appell’s functions and , Horn hypergeometric functions , and , the Lauricellas triple functions , and , and the three variables Exton’s functions , and .
Theorem 1. The following integral representations hold true:
Proof. To prove the relation (18) confirmed in Theorem 1, let denotes its right-hand side. Then, from the definition of Appell’s function in (13), we getBy applying the following integral representation of the beta function (see, e.g., [17], p. 10, (20))in (21), we obtainNow, using the well-known beta function (see, e.g., [17, 21])in (16), we get the required result. Using the same manner, we find the identities (19) and (20).
Using a similar demonstration as the previous proof, we can give the following theorems.
Theorem 2. The next integrals are true as representations of :
Theorem 3. The next integrals are true as representations of :
Theorem 4. The next integrals are true as representations of :
Theorem 5. The next integrals are true as representations of :
Theorem 6. The next integrals are true as representations of :
Theorem 7. The next integrals are true as representations of :
Theorem 8. The next integrals are true as representations of :
Theorem 9. The next integrals are true as representations of :
Theorem 10. The next integrals are true as representations of :
Now, we introduce some numerical examples for an integral representation to illustrate the above results. Consider the following integral representation of the Euler type.If we consider the integral representation given in Eq. (29) with particular values of and , we will get different expressions of . If we set and in (29), we obtain after simplificationsPutting in (52), we haveIts plot is in Figure 1, and in addition, the following examples hold true.
Example 1. For and in (53), we get
Example 2. If we let and in (57), we have
Example 3. By taking and in (57), we get
4. Integral Representations of Laplace Type
In the next, we present Laplace-type integrals for each function from (1) to (10) by means of the confluent hypergeometric functions and Humbert functions , and . First of all, we recall the following confluent hypergeometric functions (see [17]):
Theorem 11. For the hypergeometric functions and , we obtain
Proof. The previous integral representations from Eq. (58) to Eq. (67) were proved by substituting the series definition (57) which involved the confluent hypergeometric functions in each integral and by using the change of order of the integrals sign and the summations, in addition, by the help of the following known gamma integral formula [21]:Consider the following integral representation of the Laplace type:If we let in (70), we have
Example 4. If we put in (70), we havewhich, for and , we obtain
Example 5. If we let and in (70), we haveFor , we obtain
Example 6. If we take in (70), we getFor , and , we obtain
5. Conclusion
This article has established several new integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables. The new integral representations are carried out through ten three-variable hypergeometric functions, namely , and . We conclude that when = 0, mixed results derived in this work will lead to the corresponding results [22] (Gauss two-variable hypergeometric functions). Similarly, in all derived integrals, if we set and , we obtained several new integral representations of the Euler type for the classical Gauss hypergeometric function. Several numerical examples were given with graphical representation to describe some of our results .
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All three authors contributed equally in writing and obtaining this article. All authors read and approved the final manuscript.