Abstract
Agarwal et al. (2021) established the extension of several fundamental contiguous relations for . Our aim in this work is to investigate several properties of differentiation formulas, differential equations, recursion relations, differential recursion relations, confluence formulas, series representations, integration formulas, and infinite summations for Horn’s hypergeometric function of three variables. Some well-known particular cases have additionally been given.
1. Introduction and Notations
The major development of the theory of hypergeometric functions was carried out by Gauss in his famous work of 1812. In [1], Horn’s functions of two variables were initiated in the years 1889 and 1931. This paper is considered as a real starting of rigor in mathematics. Several nice papers dealing with hypergeometric functions have been investigated earlier by Euler and others, but Gauss who the first who made an interesting work of the series that defines this function. Hypergeometric series are very useful in mathematics. Note that almost all of the elementary functions of mathematics are either hypergeometric, limiting cases of a hypergeometric series, or ratios of hypergeometric functions. It is well known that there exist several types of hypergeometric functions of three variables. A number of contemporary studies have been devoted to the development of the theory of hypergeometric functions of two or more variables and its important applications, for example, wireless communication theory, nonlinear differential equations, and nonlinear cubic-quintic duffing oscillators, and so on, see [2–6]. Horn introduced four families of hypergeometric functions of three variables each. The Horn functions have several applications, for example in the theory of algebraic equations, statics, operations research, and so on. Finally, the situation completely changed due to the publication of Yang’s study, whose book was published by Academic Press [7].
Motivated mainly by these works, Sharma [8] studied properties for Appell functions. Sahin and Agha [9] introduced and studied of recursion formulas for Horn functions. In the recent papers, as Bezrodnykh [2], Dhawan [10], Exton [11], Karlsson [12], Khan and Pathan [13, 14], Padmanabham [15], Sahin [16], Sahai and Verma [17], Saran [18, 19], Srivastava [20, 21], Srivastava [22], Vidunas [23], Wang [24], it has been obtained various results about properties for hypergeometric functions of three variables. In our present work, we establish some new differentiation formulas, differential equations, recursion relations, differential recursion relations, confluence formulas, series representations, integration formulas, and infinite summations for the Horn hypergeometric function of three variables by using the technique of Ibrahim [25], Ibrahim and Rakha [26], Rakha and Ibrahim [27], Rakha et al. [28], Kim et al. [29] and Brychkov and Savischenko [30–32]. Pathan et al. [33], and Shehata and Shimaa [34, 35] found 7 and 11 hypergeometric series in two variables of order two. We shall derive these series from our viewpoint. As an application, we shall show that Theorem 2.2 leads to partial differential equations, Theorem 3.1 and Theorem 3.2 lead to differential recursion formulas, Eqs. (42)-(9) lead to series representations, Theorem 6.1 and Theorem 6.2 lead to integral representations of Euler type, and Theorem 7.1 leads to infinite summations for Horn’s series of three variables.
For the purposes of our present study, we begin by recalling here the binomial identities as follows:
The Pochhammer symbol is defined by
The Horn’s function of three variables is defined by [22, 36]:
The following basic lemma (see Srivastava and Manocha ([36], pp. 100)) given below is useful.
The integral operator is given by Abul-Ez and Sayyed, and Sayyed in [12, 37]: where , , and .
2. Differentiation Formulas
We first give the differentiation formulas and partial differential equations for the function .
Theorem 2.1. The derivative formulas hold: true for , and :
Proof. Differentiating -times with respect to , -times with respect to and -times with respect to , we obtain (7).
As special cases for , we have
Theorem 2.2. Each of the partial differential equations hold: true: where
Proof. Applying the differential operators into (4) and simplifying, we obtain (9)-(10).
We note that the results mentioned in Theorem 2.1 and Theorem 2.2 are generalizations of the results in Theorem 3.3 and Theorem 3.5 in the published paper [38].
3. Contiguous Relations
Here, we establish some recursion formulas and contiguous relations for the function with respect to parameters. We start with the following theorem. The proofs are straightforward computations.
Theorem 3.1. The Horn hypergeometric function satisfies the following relations:
Proof. From (4) and (1), we rewrite in the form If we put ,, , and in (14), we have Using (14), we rewrite (15) as follows where If we put , and in (16), we get We observe that Using (20) and (19), we get Differentiating with respect to gives Using (21) and (22), we get As special cases, letting in (23), we get In (23) and , we obtain Combining of the above, we have This implies for that Using (23), we can easily obtain the formula (13).
Now, we are going to deduce some recurrence relations for , and .
Theorem 3.2. The function satisfies the following recurrence relations:
Proof. If we put , and in (15), we get Using (17), we get We can rewrite (35) in the form This implies for that Putting in (37) implies Using (7), we get (29). Similarly, if we take , and in (16), we get (30) and (31). Again, if we take , and in (16), we get (32) and (33).
4. Confluence Formulas
The following confluence formulas follow by using the equation (4) via a direct computation.
We recall the relation as follows:
Expanding the expressions , and by applying the binomial theorem (i.e., the expansion of a function), we get the transformation formula:
5. Series Representations
Here, we derive the series representations for .
Using (4) and the summation with respect to , or , we get
Similarly, after evaluating the finite sums, it follows that
For special case, by taking , we get
Thus, we have shown that the above series (42)-(44) converge in the indicated regions.
6. Integration Formulas
We present here the integral representations and evaluation of integrals for the function .
Theorem 6.1. For , the integration formulas for Horn’s function :
Proof. With the help of the integral operator , we get the formula Applying the integral operator , we have Iterating this integral operator on for -times, we get (47). Similarly, we obtain (48) and (49).
Theorem 6.2. The integral representations for Horn’s function hold: where
Proof. If is a non-negative integer and using the integral definition of the Beta function, we have Therefore, The equation (54) can be proven as the proof of equation (52).
7. Infinite Summations for
Here, we give the infinite summations for Horn’s function .
Using the binomial theorem [36], we get
Theorem 7.1. For , the infinite summation for Horn’s function :
Proof. With the aid of (4) and (57), we directly obtain the results (58)-(62).
8. Conclusions and General Remarks
Our paper is generally based on the extension of Horn’s hypergeometric functions of two variables. A few specializations relevant to the present discussion have also been derived from results of papers [33–35, 38]. We focused on the generalization of the Horn’s function of three variables and presented some partial differential equations, differential recursion formulas, series representations, integral representations and infinite summations.
In Yang’s study, numerous special functions are summarized, and their properties are detected, which can be utilized for applications immediately, and readers can create a new special function for their own practical applications.
To be complemented, Yang’s study [7] is very useful for physical and engineering applications. It can attainable to all students, researchers, physicists, and engineers without the necessity of a profound knowledge of mathematics. Any reader can be researched for their beloved shells in the field of special functions.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
Al authors approve and are agree with the final version. .
Funding
This work does not receive any external funding.