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Certain Generating Functions Involving Some Hypergeometric Series of Four Variables by Means of Operational Representations

Jihad A. Younis, Maged G. Bin-Saad, Kottakkaran S. Nisar*

Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Saudi Arabia

Abstract: The main aim of this present paper is to present certain generating functions of some hypergeometric functions in four variables by using the integral and symbolic representations for these quadruple functions. A few interesting special cases have also been considered.

Keywords: Laplace integrals, symbolic representations, quadruple hypergeometric functions, generating functions

Received October 06, 2020; Revised November 07, 2020; Accepted November 15, 2020

Citation: Jihad A. Younis, Maged G. Bin-Saad, Kottakkaran S. Nisar, Certain Generating Functions Involving Some Hypergeometric Series of Four Variables by Means of Operational Representations. Turkish Journal of Analysis and Number Theory, 8 (6). 97-106. January 2020. doi:10.12691/tjant-8-6-1

1. Introduction

Several families of generating functions have been established in diverse ways. These are playing important roles in the theory of special functions of applied mathematics and mathematical physics. One can refer the extensive work of Srivastava and Manocha [1] for a systematic introduction to, and several interesting and useful applications of the various methods of obtaining linear, bilinear, bilateral or mixed multilateral generating functions for a fairly wide variety of sequences of hypergeometric functions and polynomials in one, two or more variables, among much abundant literature. In fact, a remarkable large number of generating functions involving a variety of hypergeometric functions have been developed by many authors such as [2,3,4,5,6].

Bin-Saad and Younis introduced in [7] thirty new quadruple hypergeometric series, ten of them defined below

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

[picture] | (5)

[picture] | (6)

[picture] | (7)

[picture] | (8)

[picture] | (9)

[picture] | (10)

Here is the Pochhammer symbol which defined as

with the assumed

It may be recalled the Laplace integral representations of the above functions, see e.g. [8,9,10,11] respectively, as below

[picture] | (11)

[picture] | (12)

[picture] | (13)

[picture] | (14)

[picture] | (15)

[picture] | (16)

[picture] | (17)

[picture] | (18)

[picture] | (19)

[picture] | (20)

where and denote the confluent hypergeometric functions [12].

The aim of this paper is to investigate the various generating functions for the quadruple hypergeometric functions In Section 2, our main generating functions are obtained with the help of integral representations of Laplace-type. In Section 3, we introduce symbolic representations for the functions. The aim of Section 4 is to use symbolic representations to derive a number of generating functions.

2. Generating Functions via Laplace Integral Representations

Here, in this Section, we used the Laplace integral representations for the hypergeometric series of four variables defined in pervious Section to determine generating functions which are listed in the form of following relations (21) to (34):

[picture] | (21)

[picture] | (22)

[picture] | (23)

[picture] | (24)

[picture] | (25)

[picture] | (26)

[picture] | (27)

[picture] | (28)

[picture] | (29)

[picture] | (30)

[picture] | (31)

[picture] | (32)

[picture] | (33)

[picture] | (34)

where is the Horn function defined in [12], and are the Lauricella functions defined in [13], is the Srivastava function defined in [12]. and are the Exton functions defined in [14] and is the Sharma and Parihar function defined in [15].

Proof. To prove (21), for convenience and simplicity, by denoting the left-hand side of (21) with and using (11), we have

Using the Kummer's transformation (see [16])

after a little simplification, we get

Using the following well-known formula (see [16,17]) after a little simplification, we easily arrive at the right-hand side of (21). This completes the proof of (21). The proof of the relations (22) to (34) runs in same way.

If we take in (32) , we shall obtain the following relation:

which, for yields the well-known result (see [12]).

By setting in (22) and (33), we have the following generating relations:

where is quadruple hypergeometric series defined by Bin-Saad and Younis (see [9] and [7]).

which, for we obtain new connection between Srivastava function and Sharma and Parihar function

3. Symbolic Representations

In this section, we consider the symbolic images for the quadruple functions (1) - (10). Our results presented here are obtained with the help of the following operator formulas:

[picture] | (35)

[picture] | (36)

where the operators and are the derivative operator and the inverse of the derivative, respectively, (see [18]). Now, we introduce certain operational representations involving quadruple functions as below:

[picture] | (37)

[picture] | (38)

[picture] | (39)

[picture] | (40)

[picture] | (41)

[picture] | (42)

[picture] | (43)

[picture] | (44)

[picture] | (45)

[picture] | (46)

where is the exponential expansion defined by (see [16])

[picture] | (47)

Proof. To prove formula (37), denote, for convenience, the left-hand side of assertion (37) by Then by the exponential expansion (47) and binomial theorem, we have

Using (35) and (36) and in view of the definition (1), we obtain the desired result (37). The proofs of formulas (38) to (46) run parallel to that of formula (37), so are skipped details.

If we take in (40) and (41), we get operational representations involving Srivastava's function (see [12]). Relation (44) with yield the Bin-Saad's and Maisoon's results [19].

4. Generating Functions via Symbolic Representations

In this section, we establish some generating relations by using the symbolic images obtained in previous section. Following are these generating relations:

[picture] | (48)

[picture] | (49)

[picture] | (50)

[picture] | (51)

[picture] | (52)

[picture] | (53)

[picture] | (54)

[picture] | (55)

[picture] | (56)

[picture] | (57)

Proof. To prove relation (48), taking in (37) and multiplying both the sides with we obtain

and then taking the double sum of both sides we get

Now, by using the definition (47), then, after some simplification gives the result. In a similar manner, one can prove the relations (49) to (57).

If we choose in (54) and (57), and simplifying, we have the following new generating functions:

which, for we have generating functions involving Appell's series [12]

which, for after a little simplification, we have interesting result

where is the Gaussian hypergeometric function (see [12]).

5. Conclusion

Based on the integral and operational representations for the hypergeometric functions of four variables defined in (1) to (10), we established several generating functions for these quadruple functions. Some particular cases and the consequences of our main results are also considered. We concluded this investigation by remarking that the scheme suggested in the derivation of the results can be applied to find other new generating functions for other quadruple hypergeometric functions and study their special cases.

Conflict of Interest

The authors have no competing interests.

References

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[2]   | Agarwal, P., Chand, M and Purohit, S. D., A note on generating functions involving the generalized Gauss hypergeometric function. National Acad. Sci. Lett., 3, 457-459, 2014.
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[13]   | Lauricella, G., Sull funzioni ipergeometric a pi variabili. Rend. Cric. Mat. Palermo, 7, 111-158, 1893.
[14]   | Exton, H., Hypergeometric functions of three variables. J. Indian Acad. Math., 4, 113-119, 1982.
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[16]   | Erdélyi, A., Magnus, W., Oberhettinger, F and Tricomi, F. G., Higher Transcendental Functions. Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
[17]   | Rainville, E. D., Special Functions. The Macmillan Company: New York, NY, USA, 1960; Reprinted by Chelsea Publishing Company, Bronx, NY, USA, 1971.
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[19]   | Bin-Saad, M. G and Hussein, M. A., Operational images and relations of two and three variable hypergeometric series. J. Progr. Res. Math., 2, 39-46, 2015.