# Some Summation Formulas for the Generalized Kampé de Fériet Function

Authors: Ashish Verma; Jihad Younis; Vikash Kumar Pandey; Hassen Aydi
Journal: Mathematical Problems in Engineering
 (2021)

---

## 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 inn variables and four generalized Lauricella functions are obtained from the finite summation formulas for the GKDFS.


---

## Body

## 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 F1, F2, and F3. 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 n variables are given.The multivariable generalization of Kampé de Fériet function is given as [2, 3](1)Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=∑s1,…,sn=0∞∧s1,e,sn∏i=1nτisisi!,where(2)∧s1,…,sn=∏j=1pajs1+⋯+sn∏j=1μ1bj1s1…∏j=1μnbjnsn∏j=1lαjs1+⋯+sn∏j=1υ1ωj1s1…∏j=1υnωjnsn,and, for convergence of (1), 1+l+υu−p−μu≥0 for u=1,…,n. The equality is satisfied if, in addition, either (p>l and τ11/p−l+⋯+τn1/p−l<1), or p≤landmaxτ1,…,τn<1.Next, we give the definition of the derivative operator [3](3)Dτfτ=υh⟶0fτ+h−fτh,provided f is differentiable at τ. Moreover, Dτnfτ=DτDτn−1fτ, n=0,1,2,….From now, we consider some abbreviated notations:(4)θp+k=θ1+k,…,θp+k,θpi+k=θ1+k,…,θi−1+k,θi+1+k,…,θp+k,i=1,…,p,ξμtt+k=ξ1t+k,……,ξμtt+k,ξμtt,i+k=ξ1t+k,…,ξi−1t+k,ξi+1t+k,…,ξμtt+k,t=1,…,n,1≤i≤μt,θk=∏j=1pθjk,θik=∏j=1,j≠ipθjk,ξtk=∏j=1μtξjtk,ξt,ik=∏j=1,j≠iμtξjtk,t=1,…,n,where θjk corresponds to Pochhammer symbol [16].


## 2. FSFs of GKDFS by Derivative Operator

The FSFs of GKDFS follow by using a derivative operator. Therth derivative on τ1 of GKDFS is obtained as follows:(5)Dτ1rFl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=arb1rαrω1rFl:υ1;…;υnp:μ1;…;μnap+r:bμ11+r;bμ22;…;bμnn;αl+r:ωυ11+r;ωυ22;…;ωυnn;τ1,…,τn.Due to the Leibnitz formula(6)Dτ1rfτ1gτ1=∑k=0rrkDτ1r−kfτ1Dτ1kgτ1,and (5), the following FSFs of GKDFS follow.Theorem 1.
We have the following FSFs of GKDFS:(7)∑k=0rrkaikb1kαkω1kτ1kFl:υ1;…;υnp:μ1;…;μnap+k:bμ11+k;bμ22;…;bμnn;αl+k:ωυ11+k;ωυ22;…;ωυnn;τ1,…,τn=Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:ai+r,bμ11;bμ22;…;bμnn;αl:ai,ωυ11;ωυ22;…;ωυnn;τ1,…,τn,with i=1,…,p;(8)∑k=0rrkakb1,irαkω1rτ1kFl:υ1;…;υnp:μ1;…;μnap+k:bμ11+k;bμ22;…;bμnn;αl+k:ωυ11+k;ωυ22;…;ωυnn;τ1,…,τn=Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:ai1+r,bμ11,i;bμ22;…;bμnn;αl:ωυ11;ωυ22;…;ωυnn;τ1,…,τn,with i=1,…,μ1.Proof.
We prove the identity (7). Using the definition of GKDFS and the Leibnitz formula for differentiation of a product of two functions, one writes(9)Dτ1rτ1ai+r−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=∑k=0rrkDτ1r−kτ1ai+r−1Dτ1kFl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=airτ1ai−1∑k=0rrkaikb1kαkω1kτ1k×Fl:υ1;…;υnp:μ1;…;μnap+k:bμ11+k;bμ22;…;bμnn;αl+k:ωυ11+k;ωυ22;…;ωυnn;τ1,…,τn.
Here, we used (5) and a simplification in the second equality. Again, combine τ1ai+r−1 with the variable τ1 in the GKDFS and put the derivative operator r-times on τ1 to get the following:(10)Dτ1rτ1ai+r−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=∑ς1,…,ςn=0∞∧ς1,…,ςnai+ς1rτ1ai−1∏i=1nτiςiςi!=airτ1ai−1Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:ai+r,bμ11;bμ22;…;bμnn;αl:ai,ωυ11;ωυ22;…;ωυnn;τ1,…,τn.
Formula (7) follows clearly. The proof of (8) is done similarly.Theorem 2.
We have the following FSF of GKDFS:(11)∑k=0rrkakb1kωi1−rkαkω1kτ1kFl:υ1;…;υnp:μ1;…;μnap+k:bμ11+k;bμ22;…;bμnn;αl+k:ωυ11+k;ωυ22;…;ωυnn;τ1,…,τn=Fl:υ1;…;υnp:μ1;…;μnap:bμ11;bμ22;…;bμnn;αl:ωi1−r,ωυ11,i;ωυ22;…;ωυnn;τ1,…,τn,where i=1,…,υ1.Proof.
As in the proof of Theorem1, the application of the derivative operator r-times on τ1ωi1−1Fl:υ1;…;υnp:μ1;…;μnτ1,…,τn yields (11). We omit the details.Theorem 3.
We have the following FSFs of GKDFS:(12)∑k=0rrk−1k1−ωi1k2−ωi1−rkτ1kFl:υ1;…;υnp:μ1;…;μnap:bμ11;bμ22;…;bμnn;αl:ωi1−k,ωυ11,i;ωυ22;…;ωυnn;τ1,…,τn=−1rarb1rωi1−1rαrω1rFl:υ1;…;υnp:μ1;…;μnap+r:bμ11+r;bμ22;…;bμnn;αl+r:ωυ11+r;ωυ22;…;ωυnn;τ1,…,τn,(13)∑k=0rrk−1kωi1+r−1kωi1kτ1kFl:υ1;…;υnp:μ1;…;μnap:bμ11;bμ22;…;bμnn;αl:ωi1+k,ωυ11,i;ωυ22;…;ωυnn;τ1,…,τn=arb1rωi1+rrαrω1rτ1rFl:υ1;…;υnp:μ1;…;μnap+r:bμ11+r;bμ22;…;bμnn;αl+r:ωi1+2r,ωυ11,i+r;ωυ22;…;ωυnn;τ1,…,τn,where i=1,…,υ1.Proof.
We first establish (12). Due to GKDFS and the Leibnitz formula for differentiation of a product of two functions, one gets(14)Dτ1rτ11−ωi1×τ11−ωi1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=∑k=0rrkDτ1r−kτ11−ωi1Dτ1kτ1ωi1−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=∑k=0r−1r+krkωi1−1r1−ωi1k2−ωi1−rkτ1r×Fl:υ1;…;υnp:μ1;…;μnap:bμ11+k;bμ22;…;bμnn;αl:ωi1−k,ωυ11,i;ωυ22;…;ωυnn;τ1,…,τn.
The use of the derivative operatorr-times on GKDFS and the combination with the above lead to (12). The application of Dτ1r on(15)τ11−ωi1−r×τ1ωi1+r−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11+r;bμ22;…;bμnn;αl:ωi1+r,ωυ11,i;ωυ22;…;ωυnn;τ1,…,τn.
Formula (13) follows as in the proof of (12).Theorem 4.
We have the following FSFs of GKDFS:(16)∑k=0r−rkai−r+1kFl+1:υ1:…:υnp+1:μ1:…:μn1+k,ap:bμi1;…;bμn1;1,αl:ωυi1;…;ωυn1;1τ1,…1τ1,=ai−raiFl+1:υ1:…:υnp+1:μ1:…:μn1−ai+r,ap:bμi1;…;bμn1;1−ai,αl:ωυi1;…;ωυn1;1τ1,…1τ1,with i=1,…,p;(17)∑k=0r−rkbi1−r+1kFl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:1+k,bμ11;…;bμnn;αl:1,ωυ11;…;ωυnn;1τ1,τ2,…,τn=bi1−rbi1Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:1−bi1+r,bμ11;…;bμnn;αl:1−bi1,ωυ11;…;ωυnn;1τ1,τ2,…,τn,with i=1,…,μ1.Proof.
We first establish (16). Note that formula (17) could be done similarly. The calculation of rth derivatives on τ1 of τ1ai−1Fl:υ1;…;υnp:μ1;…;μn1/τ1,…,1/τn gives(18)Dτ1rτ1ai−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=−1r1−airτ1r+1−aiFl+1:υ1;…;υnp+1:μ1;…;μn1−ai+r,ap:bμ11;…;bμnn;1−ai,αl:ωυ11;…;ωυnn;1τ1,…,1τ1.
Due to the Leibnitz formula, one has(19)Dτ1rτ1ai−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=∑k=0rrkDτ1r−kτ1aiDτ1kτ1−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=−1r−airτ1ai−r−1∑k=0r−rkai−r+1kFl+1:υ1;…;υnp+1:μ1;…;μn1+k,ap:bμ11;…;bμnn;1,αl:ωυ11;…;ωυnn;1τ1,…,1τ1.
With the combination of the above, one has (16).Theorem 5.
The following FSFs of GKDFS are satisfied:(20)∑k=0r−rk2−ai−rkFl+1:υ1;…;υnp+1:μ1;…;μn1+k,api:bμ11;…;bμnn;1,αl:ωυ11;…;ωυnn;1τ1,…,1τ1=ai+r−1ai−1Fl:υ1;…;υnp:μ1;…;μnai+r,api:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1,with i=1,…,p;(21)∑k=0r−rk2−bi1−rkFl+1:υ1;…;υnp+1:μ1;…;μnap:1+k,bμ11+k;…;bμnn;αl:1,ωυ11+k;…;ωυnn;1τ1,τ2,…,τn=bi1+r−1bi1−1Fl:υ1;…;υnp:μ1;…;μnap:bi1+r,bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,τ2,…,τn,with i=1,…,μ1.Proof.
We multiply the L. H. S ofFl:υ1;…;υnp:μ1;…;μn1/τ1,…,1/τn series by τ11−ai×τ1−1 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:(22)∑k=0r−rk2−2rkFl+1:υ1;…;υnp+1:μ1;…;μn1+k,ap:bμ11;…;bμnn;1,αl:ωυ11;…;ωυnn;1τ1,…,1τ1=2r−1r−1Fl+1:υ1;…;υnp+1:μ1;…;μn2r,ap:bi1+r,bμ11;…;bμnn;r,αl:ωυ11;…;ωυnn;1τ1,…,1τ1,with i=1,…,p;(23)∑k=0r−rk2−2rksFl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:1+k,bμ11;…;bμnn;αl:1,ωυ11;…;ωυnn;1τ1,τ2,…,τn=2r−1r−1Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:2r,bμ11;…;bμnn;αl:r,ωυ11;…;ωυnn;1τ1,τ2,…,τn,with i=1,…,μ1.Proof.
We first present an idea to establish (22). We multiply the L. H. S of Fl:υ1;……;υnp:μ1;…;μn1/τ1,…,1/τn-series by τ11−r×τ1−1 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:(24)∑k=0r−rk1−2rkFl+1:υ1;…;υnp+1:μ1;…;μn1+k,ap:bμ11;…;bμnn;1,αl:ωυ11;…;ωυnn;1τ1,…,1τ1=2Fl+1:υ1;…;υnp+1:μ1;…;μn1+2r,ap:bi1+r,bμ11;…;bμnn;1+r,αl:ωυ11;…;ωυnn;1τ1,…,1τ1,(25)∑k=0r−rk1−2rkFl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:1+k,bμ11;…;bμnn;αl:1,ωυ11;…;ωυnn;1τ1,τ2,…,τn=2Fl:υ1+1;υ2;…;υnp:μ1+1;μ2;…;μnap:1+2r,bi1+r,bμ11;…;bμnn;αl:1+r,ωυ11;…;ωυnn;1τ1,τ2,…,τn.Proof.
In view of the derivative operatorr-times, one finds(26)Dτ1rτ1−r−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=∑ς1,…,ςn=0∞∧ς1,…,ςnτ1−ς1−ς2⋯−ςn−2r−1∏i=1nςi!−1rς1+⋯+ςn+r+1r=−1r1+rrτ1−2r−1Fl+1:υ1;…;υnp+1:μ1;…;μn1+2r,ap:bi1+r,bμ11;…;bμnn;1+r,αl:ωυ11;…;ωυnn;1τ1,…,1τ1.
The application of the Leibnitz formula implies that(27)Dτ1rτ1−r−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=∑k=0rrkDτ1r−kτ1−rDτ1kτ1−1Fl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;1τ1,…,1τ1=−1rrrτ1−2r−1∑k=0r−rk1−2rkFl+1:υ1;…;υnp+1:μ1;…;μn1+k,ap:bi1+r,bμ11;…;bμnn;1,αl:ωυ11;…;ωυnn;1τ1,…,1τ1.
We combine the above to get (24). Identity (25) is established similarly.Theorem 8.
The following FSFs of GKDFS are verified:(28)∑k=0rrk−1k−rk1+ai−rkFl+α:υ1;…;υnp+α:μ1;…;μn1+rα,…,α+rα,ap:bμ11;…;bμnn;1+r−kα,…,α+r−kα,αl:ωυ11;…;ωυnn;τ1α,…,τ1α=−1r1+air−airFl+α:υ1;…;υnp+α:μ1;…;μn1+ai+rα,…,α+ai+rα,ap:bμ11;…;bμnn;1+aiα,…,α+aiα,αl:ωυ11;…;ωυnn;τ1α,…,τ1α,α≥2,with i=1,…,p;(29)∑k=0rrk−1k−rk1+bi1−rkFl:υ1+α;υ2;…;υnp:μ1+α;μ2;…;μnap:1+rα,…,α+rα:bμ11;…;bμnn;αl:1+r−kα,…,α+r−kα:ωυ11;…;ωυnn;τ1α,τ2,…,τn=−1r1+bi1r−bi1rFl:υ1+α;υ2;…;υnp:μ1+α;μ2;…;μnap:1+bi1+rα,…,α+bi1+rα,bμ11;…;bμnn;αl:1+bi1α,…,α+bi1α,ωυ11;…;ωυnn;τ1α,τ2,…,τn,α≥2,with i=1,…,μ1.Proof.
Firstly, we establish (28). The derivative operator r-times leads to(30)Dτ1rτ1ai+rFl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1α,…,τ1α=∑ς1,…,ςn=0∞∧ς1,…,ςnτ1ας1+⋯+ςn+ai∏i=1nςi!ας1+⋯+ςn+ai+r!ας1+⋯+ςn+ai!=τ1ai∑ς1,…,ςn=0∞∧ς1,…,ςnτ1ας1+⋯+ςn∏i=1nςiai+r!1+ai+r/α,…,α+ai+r/ας1+⋯+ςnai!1+ai/α,…,α+ai/ας1+⋯+ςn=τ1ai1+airFl+α:υ1;…;υnp+α:μ1;…;μn1+ai+rα,…,α+ai+rα,ap:bμ11;…;bμnn;1+aiα,…,α+aiα,αl:ωυ11;…;ωυnn;τ1α,…,τ1α.
Again, Leibnitz formula implies that(31)Dτ1rτ1ai+rFl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1α,…,τ1α=∑k=0rrkDτ1r−kτ1aiDτ1kτ1rFl:υ1;…;υnp:μ1;…;μnap:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1α,…,τ1α=∑k=0rrk−1r+k−air−rk1+ai−rkτ1ai×Fl+α:υ1;…;υnp+α:μ1;…;μn1+rα,…,α+rα,ap:bμ11;…;bμnn;1+r−kα,…,α+r−kα,αl:ωυ11;…;ωυnn;τ1α,…,τ1α.
Equating 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:(32)∑k=0r−1kai+1kai+1−ai−r+1kFl:υ1;…;υnp:μ1;…;μnai+1+k,api+1:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=airai−ai+1rFl:υ1;…;υnp:μ1;…;μnai+r,api:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn,with i=1,…,p−1;(33)∑k=0r−1kbi+11kbi+11−bi1−r+1kFl:υ1;…;υnp:μ1;…;μnap:bi+11+k,bμ11,i;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn=bi1rbi1−bi+11rFl:υ1;…;υnp:μ1;…;μnap:bi1+r,bμ11,i;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn,with i=1,…,μ1−1.Proof.
In view of the definition of GKDFS, the L. H. S of (32) may be written as(34)∑ς1,…,ςn=0∞∧ς1,…,ςn∏i=1nτiςiςi!2F1−r,ai+1+r;ai+1−ai−r+1;1=airai−ai+1rFl:υ1;…;υnp:μ1;…;μnai+r,api:bμ11;…;bμnn;αl:ωυ11;…;ωυnn;τ1,…,τn.
The application of theorem of Vandermonde,(35)F21−r,a;b;1=b−arbr,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 n variables Φ2n, Ψ2n, ΦDn, Ξ1n, and Φ3n [2]. For instance, characterizing the parameters in (8), we got the finite summation identities for FBn and Ξ1n:(36)∑k=0rrkξ1kckτ1kFBnθ1+k,ξ1+k,ξ2,…,ξn;c+k;τ1,…,τn=FBnθ1+r,ξ1,…,ξn;c;τ1,…,τ1;∑k=0rrkξ1kckτ1kΞ1nθ1+k,θ2,…,θn,ξ1+k,ξ2,…,ξn−1;c+k;τ1,…,τn=Ξ1nθ1+r,ξ1,…,ξn−1;c;τ1,…,τ1.Also, characterizing the parameters in (24), we established the following summation identity for FDn and ΦDn:(37)∑k=0r−rk1−2rkFDn1+r,ξ1,…,ξn;1;1τ1,…,1τ1=2FDn1+2r,ξ1,…,ξn;1+r;1τ1,…,1τ1;∑k=0r−rk1−2rkΦDn1+r,ξ1,…,ξn−1;1;1τ1,…,1τ1=2ΦDn1+2r,ξ1,…,ξn−1;1+r;1τ1,…,1τ1.



---

*Source: 2861820-2021-09-15.xml*