Abstract

The aim of this manuscript is to establish several finite summation formulas (FSFs) for the generalized Kampé de Fériet series (GKDFS). Moreover, the particular result for confluent forms of Lauricella series in variables and four generalized Lauricella functions are obtained from the finite summation formulas for the GKDFS.

1. Introduction

Special functions are essential tools in several equations arising in natural science. The hypergeometric series and its generalizations are appeared in many mathematical problems and their applications. The theory of hypergeometric functions in many variables by the fact that the solutions of partial differential equations appearing in several applied problems of mathematical physics has been presented in terms of such hypergeometric functions [1–4].

Since 2012, Brychkov and Saad [5–8] have obtained many finite summation formulas of Appell’s functions , , and . Later, Wang established some infinite summation formulas of double hypergeometric functions [9]. In 2016, Wang and Chen [10] derived FSFs of double hypergeometric functions involving some summation theorems. In 2019, Sahai and Verma [11] gave FSFs for the Srivastava’s general triple hypergeometric function [12]. For instance, works of Lauricella functions [13] and Srivastava’s triple hypergeometric functions [14, 15] have been provided. These works generalized and unified several results in [10] for the three-variable hypergeometric function. In view of the abovementioned works, our motivation is to present here several FSFs for the GKDFS. Also, some particular cases yielding to FSFs for four generalized Lauricella functions and confluent forms of Lauricella series in variables are given.

The multivariable generalization of Kampé de Fériet function is given as [2, 3]whereand, for convergence of (1), for . The equality is satisfied if, in addition, either ( and ), or .

Next, we give the definition of the derivative operator [3]provided is differentiable at . Moreover, , .

From now, we consider some abbreviated notations:where corresponds to Pochhammer symbol [16].

2. FSFs of GKDFS by Derivative Operator

The FSFs of GKDFS follow by using a derivative operator. The th derivative on of GKDFS is obtained as follows:

Due to the Leibnitz formulaand (5), the following FSFs of GKDFS follow.

Theorem 1. We have the following FSFs of GKDFS:with ;with .

Proof. We prove the identity (7). Using the definition of GKDFS and the Leibnitz formula for differentiation of a product of two functions, one writesHere, we used (5) and a simplification in the second equality. Again, combine with the variable in the GKDFS and put the derivative operator -times on to get the following:Formula (7) follows clearly. The proof of (8) is done similarly.

Theorem 2. We have the following FSF of GKDFS:where .

Proof. As in the proof of Theorem 1, the application of the derivative operator -times on yields (11). We omit the details.

Theorem 3. We have the following FSFs of GKDFS:where .

Proof. We first establish (12). Due to GKDFS and the Leibnitz formula for differentiation of a product of two functions, one getsThe use of the derivative operator -times on GKDFS and the combination with the above lead to (12). The application of onFormula (13) follows as in the proof of (12).

Theorem 4. We have the following FSFs of GKDFS:with ;with .

Proof. We first establish (16). Note that formula (17) could be done similarly. The calculation of th derivatives on of givesDue to the Leibnitz formula, one hasWith the combination of the above, one has (16).

Theorem 5. The following FSFs of GKDFS are satisfied:with ;with .

Proof. We multiply the L. H. S of series by and we use the derivative operator as in (16) to prove (20). The proof of formula (21) is done similarly.

Theorem 6. The following FSFs of GKDFS are verified:with ;with .

Proof. We first present an idea to establish (22). We multiply the L. H. S of -series by and we apply the derivative operator on as in the proof of (16) to obtain (22). Formula (23) may be done in a similar strategy.

Theorem 7. The following FSFs of GKDFS are verified:

Proof. In view of the derivative operator -times, one findsThe application of the Leibnitz formula implies thatWe combine the above to get (24). Identity (25) is established similarly.

Theorem 8. The following FSFs of GKDFS are verified:with ;with .

Proof. Firstly, we establish (28). The derivative operator -times leads toAgain, Leibnitz formula implies thatEquating the above two identities yields (28). The other equality (29) can be established similarly.

3. FSFs of GKDFS by Rearrangement

Theorem 9. The following FSFs of GKDFS are verified:with ;with .

Proof. In view of the definition of GKDFS, the L. H. S of (32) may be written asThe application of theorem of Vandermonde,in the above equation gives (32). Result (33) could be obtained in a similar strategy.

4. Conclusion

We gave many finite summation identities using the GKDFS. By specializing the parameters in GKDFS, we obtain summation formulas for the generalized Lauricella functions [2, 3], as well as confluent forms of Lauricella series in variables , , , , and [2]. For instance, characterizing the parameters in (8), we got the finite summation identities for and :

Also, characterizing the parameters in (24), we established the following summation identity for and :

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.