# On the Kampé de Fériet Hypergeometric Matrix Function

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

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## Abstract

In this study, we derive recursion formulas for the Kampé de Fériet hypergeometric matrix function. We also obtain some finite matrix and infinite matrix summation formulas for the Kampé de Fériet hypergeometric matrix function.


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## Body

## 1. Introduction

The theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic computation and engineering problems usually recognize the majority of special functions. Recently, there has been a surge in the study of recursion formulas for multivariable hypergeometric functions. Recursion formulas for the Appell functionF2 have been investigated by Opps et al. [1], followed by Wang [2], who presented the recursion relations for all Appell functions. Furthermore, recursion formulas for variant multivariable hypergeometric functions were presented in [3–6]. One can refer to various sources [7, 8] for the in-depth study of the hypergeometric functions for several variables.The theory of generalized matrix special functions has witnessed a rather significant evolution during the last two decades. The reasons of interest have a manifold motivation. Restricting ourselves to the applicative field, we note that for some physical problems, the use of new classes of matrix special functions provided solutions hardly achievable with conventional analytical and numerical means. Special matrix functions appear in the literature related to statistics [9], Lie theory [10], and more recently in connection with the matrix version of Laguerre, Hermite, and Legendre differential equations and the corresponding polynomial families [11–13]. In [14], recursion formulas and matrix summation formulas for Srivastava’s triple hypergeometric matrix functions are obtained.The study is organized in the following manner. In Section2, we list basic definitions that are needed in the sequel. In Section 3, we obtain recursion formulas for the Kampé de Fériet hypergeometric matrix function (its abbreviation is K de FHMF). In Section 4, we present finite matrix summation formulas for the (K de FHMF) by applying a derivative operator. Finally, in Section 5, we establish infinite matrix summation formulas for the (K de FHMF).


## 2. Preliminaries

Letℂr×r be the vector space of r square matrices with complex entries. For any matrix H∈ℂr×r, its spectrum σH is the set of eigenvalues of H. H in ℂr×r is called a positive stable matrix if ℜλ>0 for all λ∈σH.The reciprocal gamma functionΓ−1θ=1/Γθ is an entire function of the complex variable θ. The image of Γ−1θ acting on H, denoted by Γ−1H, is a well-defined matrix. If H+ℓI is invertible for all integers ℓ≥0, then the reciprocal gamma function [15] is defined by Γ−1H=HℓΓ−1H+ℓI, where Hℓ is the shifted factorial matrix function for H∈ℂr×r given as [16](1)Hℓ=I,ℓ=0,HH+I,…,H+ℓ−1I,ℓ≥1.I being the r-square identity matrix. If H∈ℂr×r is a positive stable matrix and ℓ≥1, then by [15], we have ΓH=limℓ⟶∞ℓ−1!Hℓ−1ℓH.The Gauss hypergeometric matrix function [16] is defined by(2)F21A,B;C;x=∑n=0∞AnBnCn−1n!xn,for matrices A, B, and C in ℂr×r, such that C+kI is invertible for all k≥0 and x≤1.The Appell matrix functions are defined by(3)F1A,B,B′;C;x,y=∑m,n=0∞Am+nBmB′nCm+n−1m!n!xmyn,(4)F2A,B,B′;C,C′;x,y=∑m,n=0∞Am+nBmB′nCm−1C′n−1m!n!xmyn,(5)F3A,A′,B,B′;C;x,y=∑m,n=0∞AmA′nBmB′nCm+n−1m!n!xmyn,(6)F4A,B;C,C′;x,y=∑m,n=0∞Am+nBm+nCm−1C′n−1m!n!xmyn,where A, A′, B, B′, C, and C′ are the positive stable matrices in ℂr×r, so that C+kI and C′+kI are invertible for each integer k≥0. For regions of convergence of equations (3)–(6), see [17–19].The Kampé de Fériet hypergeometric matrix function is given as [18, 19](7)Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y=∑m,n≥0∏i=1m1Aim+n∏i=1n1Bim∏i=1n1′Cin∏i=1m2Dim+n−1∏i=1n2Eim−1∏i=1n2′Fin−1xmynm!n!,where A abbreviates the sequence of matrices A1,…,Am1, and Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that Di+kI, Ei+kI and Fi+kI are invertible for all integers k≥0.Next, we recall the definition of the derivative operator(8)Dyfy=limh⟶0fy+h−fyh,provided f is differentiable at y. Also, Dykfy=DyDyk−1fy for k=0,1,2,….In the whole study,I is the identity matrix and s is a nonnegative integer. In the sequel, consider(9)A+sI=A1+sI,A2+sI,…,Am1+sI,Ai=A1,…,Ai−1,Ai+1,…,Am1,Ai+sI=A1+sI,…,Ai−1+sI,Ai+1+sI,…,Am1+sI.Also, we denote(10)A+kIs=∏i=1m1Ai+kIs,A+kIs−1=∏i=1m1Ai+kIs−1,Aj+kIs=∏i=1,i≠jm1Ai+kIs,Aj+kIs−1=∏i=1,i≠jm1Ai+kIs−1.


## 3. Recursion Formulas for the Kampé de Fériet Hypergeometric Matrix Function (K de FHMF)

In this section, we obtain the recursion formulas for the (K de FHMF).Theorem 1.
LetAi+sI,i=1,…,m1 be invertible for each integer s≥0. Then, the following recursion formula holds true for the (K de FHMF):(11)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+sI:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xAiB∑k=1sFm2;n2,n2′m1;n1,n1′;D+I:E+I,FAi+I,Ai+kI:B+I,Cx,yD−1E−1+yAiC∑k=1sFm2;n2,n2′m1;n1,n1′;D+I:E,F+IAi+I,Ai+kI:B,C+Ix,yD−1F−1.Also, ifAi−kI is invertible for integers k≤s, then(12)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai−sI:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y−xAiB∑k=0s−1Fm2;n2,n2′m1;n1,n1′;D+I:E+I,FAi+I,Ai−kI:B+I,Cx,yD−1E−1−yAiC∑k=0s−1Fm2;n2,n2′m1;n1,n1′;D+I:E,F+IAi+I,Ai−kI:B,C+Ix,yD−1F−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiAj=AjAi; AiBj=BjAi; AiCj=CjAi; BiCj=CjBi; FiEj=EjFi; FjDi=DiFj; and DiEj=EjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.Proof.
In view of equation (7) and the fact that(13)Ai+Im+n=Ai−1Aim+nAi+mI+nI,we get the following contiguous matrix relation:(14)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+I:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xAiBFm2;n2,n2′m1;n1,n1′;D+I:E+I,FA+I:B+I,Cx,yD−1E−1+yAiCFm2;n2,n2′m1;n1,n1′;D+I:E,F+IA+I:B,C+Ix,yD−1F−1.
ReplacingAi with Ai+I in equation (14), we have the following contiguous matrix relation:(15)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+2I:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xAiB∑k=12Fm2;n2,n2′m1;n1,n1′;D+I:E+I,FAi+I,Ai+kI:B+I,Cx,yD−1E−1+yAiC∑k=12Fm2;n2,n2′m1;n1,n1′;D+I:E,F+IAi+I,Ai+kI:B,C+Ix,yD−1F−1.
Iterating this processs-times, we get equation (11). For the proof of equation (12), replace the matrix Ai with Ai−I in equation (14). As Ai−I is invertible, we have(16)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai−I:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y−xAiBFm2;n2,n2′m1;n1,n1′;D+I:E+I,FAi+I,Ai:B+I,Cx,yD−1E−1−yAiCFm2;n2,n2′m1;n1,n1′;D+I:E,F+IAi+I,Ai:B,C+Ix,yD−1F−1.
Iteratively, we get equation (12).
Using contiguous matrix relations equations (14) and (16), we get the following forms of the recursion formulas for the (K de FHMF).Theorem 2.
LetAi+sI,i=1,…,m1 be invertible for each integer s≥0. Then, the following recursion formula holds true for the (K de FHMF):(17)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+sI:B,Cx,y=∑k1+k2≤ssk1,k2Aik1+k2Bk1Ck2×xk1yk2Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1.Also, ifAi−kI is invertible for integers k≤s, one writes(18)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai−sI:B,Cx,y=∑k1+k2≤ssk1,k2Aik1+k2Bk1Ck2×−xk1−yk2Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IAi+k1+k2I,Ai:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiAj=AjAi; AiBj=BjAi; AiCj=CjAi; BiCj=CjBi; FiEj=EjFi; FjDi=DiFj; and DiEj=EjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.Proof.
We prove equation (17) by applying a mathematical induction on s. For s=1, the result equation (17) is true due to equation (14). Assume equation (17) is true for s=t, that is,(19)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+tI:B,Cx,y=∑k1+k2≤ttk1,k2Aik1+k2Bk1Ck2×xk1yk2Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1.
ReplacingAi with Ai+I in equation (19) and using the contiguous matrix relation equation (14), we get(20)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+tI+I:B,Cx,y=∑k1+k2≤ttk1,k2Aik1+k2Bk1Ck2xk1yk2Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1+∑k1+k2≤t+1tk1−1,k2Aik1+k2Bk1Ck2xk1yk2×Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1+∑k1+k2≤t+1tk1,k2−1Aik1+k2Bk1Ck2xk1yk2×Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1.
Applying the known relationnk+nk−1=n+1k and nk=0 (for k>n or k<0), the above identity can be reduced to the following result:(21)Fm2;n2,n2′m1;n1,n1′;D:E,FAi,Ai+tI+I:B,Cx,y=∑k1+k2≤t+1t+1k1,k2Aik1+k2Bk1Ck2×xk1yk2Fm2;n2,n2′m1;n1,n1′;D+k1+k2I:E+k1I,F+k2IA+k1+k2I:B+k1I,C+k2Ix,yDk1+k2−1Ek1−1Fk2−1.
This establishes equation (17) for s=t+1. Hence, the result equation (17) is true for all values of s. The second recursion formula (18) is proved in a similar manner.
Now, we present the recursion formulas for matricesBi and Ci of the (K de FHMF). The proofs of the following results are omitted.Theorem 3.
LetBi+sI,i=1,…,n1 be invertible for each integer s≥0. Then, the following recursion formula is satisfied for the (K de FHMF):(22)Fm2;n2,n2′m1;n1,n1′;D:E,FA:Bi,Bi+sI,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xABi∑k=1sFm2;n2,n2′m1;n1,n1′;D+I:E+I,FA+I:Bi+I,Bi+kI,Cx,yD−1E−1.In addition, ifBi−kI is invertible for integers k≤s, one gets(23)Fm2;n2,n2′m1;n1,n1′;D:E,FA:Bi,Bi−sI,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y−xABi∑k=0s−1Fm2;n2,n2′m1;n1,n1′;D+I:E+I,FA+I:Bi+I,Bi−kI,Cx,yD−1E−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiBj=BjAi; BiBj=BjBi; FjDi=DiFj; DiEj=EjDi; and FiEj=EjFi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.Theorem 4.
LetBi+sI,i=1,…,n1 be invertible for each integer s≥0. Then, the following recursion formula is verified for the (K de FHMF):(24)Fm2;n2,n2′m1;n1,n1′;D:E,FA:Bi,Bi+sI,Cx,y=∑k=0sskAkBikxk×Fm2;n2,n2′m1;n1,n1′;D+kI:E+kI,FA+kI:B+kI,Cx,yDk−1Ek−1.Also, ifBi−kI is invertible for integers k≤s, we get(25)Fm2;n2,n2′m1;n1,n1′;D:E,FA:Bi,Bi−sI,Cx,y=∑k=0sskAkBik−xk×Fm2;n2,n2′m1;n1,n1′;D+kI:E+kI,FA+kI:Bi+kI,Bi,Cx,yDk−1Ek−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiBj=BjAi; BiBj=BjBi; FjDi=DiFj; DiEj=EjDi; and FiEj=EjFi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.The recursion formulas forFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Ci,Ci±sIx,y are obtained by replacing B↔C, E↔F, and x↔y in Theorems 3 and 4.Next, we state the recursion formulas for the matrixDi of the (K de FHMF).Theorem 5.
LetDi−sI,i=1,…,m2 be invertible for each integer s≥0. Then, the following recursion formula holds true for the (K de FHMF):(26)Fm2;n2,n2′m1;n1,n1′;Di,Di−sI:E,FA:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xAB∑k=1sFm2;n2,n2′m1;n1,n1′;Di+I,Di+2−kI:E+I,FA+I:B+I,Cx,yDi−kI−1Di−k−1I−1Di−1E−1+yAC∑k=1sFm2;n2,n2′m1;n1,n1′;Di+I,Di+2−kI:E,F+IA+I:B,C+Ix,yDi−kI−1Di−k−1I−1Di−1F−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiBj=BjAi; AiCj=CjAi; BiCj=CjBi; FiEj=EjFi; DiDj=DjDi; FjDi=DiFj; and DiEj=EjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.Proof.
Applying the definition of the (K de FHMF) and the fact that(27)Di−Im+n−1=Dim+n−11+mDi−I−1+nDi−I−1,the following contiguous matrix relation is obtained:(28)Fm2;n2,n2′m1;n1,n1′;Di,Di−I:E,FA:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xABFm2;n2,n2′m1;n1,n1′;D+I:E+I,FA+I:B+I,Cx,yDi−I−1D−1E−1+yACFm2;n2,n2′m1;n1,n1′;D+I:E,F+IA+I:B,C+Ix,yDi−I−1D−1F−1.
With the help of this contiguous matrix relation to the (K de FHMF) with the matrixDi−sI for s-times, we get equation (26).
Next, we give recursion formulas for the (K de FHMF)Ei, Fi. The proof is omitted for the following result.Theorem 6.
LetEi−sI,i=1,…,n2 be invertible for each integer s≥0. Then, the following recursion formula holds true for the (K de FHMF):(29)Fm2;n2,n2′m1;n1,n1′;D:Ei,Ei−sI,FA:B,Cx,y=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y+xAB∑k=1sFm2;n2,n2′m1;n1,n1′;D+I:Ei+I,Ei+2−kI,FA+I:B+I,Cx,yEi−kI−1Ei−k−1I−1Ei−1D−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiBj=BjAi; EiEj=EjEi; FiEj=EjFi; FjDj=DjFi; and DjEi=EiDj, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.The recursion formulas forFm2;n2,n2′m1;n1,n1′;D:E,Fi,Fi−sIA:B,Cx,y are obtained by replacing B↔C, E↔F, and x↔y in Theorem 6.


## 4. Finite Matrix Summation Formulas for the Kampé de Fériet Hypergeometric Matrix Function by a Derivative Operator

In this section, we obtain the finite matrix summation formulas for the (K de FHMF) by a derivative operator. These formulas are matrix analogues for some summation formulas of double hypergeometric functions [8]. The pth derivative on y of the (K de FHMF) is obtained as follows:(30)DypFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y=ApCpFm2;n2,n2′m1;n1,n1′;D+pI:E,F+pIA+pI:B,C+pIx,yDp−1Fp−1,where Ai, Bi, Ci, Di, Ei, and Fi are the positive stable matrices in ℂr×r, such that AiCj=CjAi; BiCj=CjBi; DiEj=EjDi; and DiFj=FjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0.By using the generalized Leibnitz formula,(31)Dypfygy=∑k=0ppkDyp−kfyDykgy,and equation (30), we derive the following finite matrix summation formulas of the (K de FHMF).Theorem 7.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that AiCj=CjAi; BiCj=CjBi; CiCj=CjCi; DiEj=EjDi; and DiFj=FjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for all integers k≥0. Then, the following finite matrix summation formulas hold for the (K de FHMF):(32)∑k=0ppkAkCikykFm2;n2,n2′m1;n1,n1′;D+kI:E,F+kIA+kI:B,C+kIx,yDk−1Fk−1=Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Ci,Ci+pIx,y.Proof.
From definition of the (K de FHMF) and the generalized Leibnitz formula for differentiation of a product of two functions, we have(33)DypyCi+pI−IFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y=∑k=0ppkDyp−kyCi+pI−IDykFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y=CipyCi−I∑k=0ppkAkCik×ykFm2;n2,n2′m1;n1,n1′;D+kI:E,F+kIA+kI:B,C+kIx,yDk−1Fk−1.
We used equation (30) and some simplification in the second equality. Next, we combine yCi+p−1I with the variable y in the (K de FHMF) and apply the derivative operator p-times on y to get the following result:(34)DypyCi+p−1IFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y=∑m,n=0∞Ci+nIp∏i=1m1Aim+n∏i=1n1Bim∏i=1n1′Cin∏i=1m2Dim+n−1∏i=1n2Eim−1∏i=1n2′Fin−1×xmyCi+n−1Im!n!=CipyCi−IFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Ci+pI,Cix,y.
Equating the above two relations leads to equation (32).Theorem 8.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that AiCj=CjAi; BiCj=CjBi; DiEj=EjDi; DiFj=FjDi; and FiFj=FjFi, and Di+kI, Ei+kI, and Fi+kI are invertible for all integers k≥0. Then, the following finite matrix summation formulas of the (K de FHMF) hold true:(35)∑k=0ppkAkCkykFm2;n2,n2′m1;n1,n1′;D+kI:E,F+kIA+kI:B,C+kIx,yFi−pIk−1Dk−1Fk−1=Fm2;n2,n2′m1;n1,n1′;D:E,Fi−pI,FiA:B,Cx,y,where Fi+k−pI is an invertible matrix for 0≤k≤p and i=1,…,n2′.Proof.
Applying the derivative operator onFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,yyFi−I, p-times, gives the formula as explained in the proof of Theorem 7. We omit the details.
Applying a derivative operator and some transformations, we can get the finite matrix summation formulas of the (K de FHMF) as follows.Theorem 9.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that AiCj=CjAi; BiCj=CjBi; DiEj=EjDi; and DiFj=FjDi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0. Then, the following finite matrix summation formulas of the (K de FHMF) hold true:(36)∑k=0rrk−1kFm2;n2,n2′m1;n1,n1′;D:E,Fi−kI,FiA:B,Cx,yI−Fik2−rI−Fik−1=−1rArCryrFm2;n2,n2′m1;n1,n1′;D+rI:E,F+rIA+rI:B,C+rIx,yFi−Ir−1Dr−1Fr−1,(37)∑k=0rrk−1kFm2;n2,n2′m1;n1,n1′;D:E,Fi+kI,FiA:B,Cx,yFi+r−1IkFik−1=ArCryrFm2;n2,n2′m1;n1,n1′;D+rI:E,Fi+2rI,Fi+rIA+rI:B,C+rIx,yDr−1Fr−1Fi+rIr−1,where 2+k−rI−Fi, Fi−kI, and Fi+k−1I is an invertible matrix for 0≤k≤r in (36); Fi+rI is an invertible matrix in equation (37), i=1,…,n2′.Proof.
We first prove identity equation (36). From the definition of the (K de FHMF) and the generalized Leibnitz formula for differentiation of a product of two functions, we obtain the following result:(38)DyrFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,yyI−Fi×yFi−I=∑k=0rrkDykFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,yyFi−IDyr−kyI−Fi=∑k=0r−1r+krkFm2;n2,n2′m1;n1,n1′;D:E,Fi−kI,FiA:B,Cx,y×Fi−IrI−Fik2I−Fi−rIk−1y−r.
Now, using the derivative operator on the (K de FHMF) forr-times directly and equating with the above equality gives equation (36) after some simplifications. Next, applying the operator Dyr on(39)Fm2;n2,n2′m1;n1,n1′;D:E,Fi+rI,FiA:B,Cx,yyI−Fi−rI×yFi+Ir−I,and proceeding as in the proof of equation (36) gives the result equation (37).


## 5. Infinite Summation Formulas for the Kampé de Fériet Hypergeometric Matrix Function

In this section, we will establish the infinite summation formulas of the (K de FHMF).Theorem 10.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0. Then, the following infinite summation formulas of the (K de FHMF) hold true:(40)∑k=0∞Aikk!tkFm2;n2,n2′m1;n1,n1′;D:E,FAi+kI,Ai:B,Cx,y=1−t−AiFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx1−t,y1−t,where i=1,…,m1;(41)∑k=0∞Bikk!tkFm2;n2,n2′m1;n1,n1′;D:E,FA:Bi+kI,Bi,Cx,y=1−t−BiFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx1−t,y,AiBj=BjAi,where i=1,…,n1.Proof:.
We shall prove equation (40). We apply the definition of the (K de FHMF) and transformation,(42)AikAi+kIm+n=Aim+nAi+m+nIk,to get that the left side of equation (40) is written as(43)∑m,n=0∞F10;−Ai+m+nIt∏i=1m1Aim+n∏i=1n1Bim∏i=1n1′Cin∏i=1m2Dim+n−1∏i=1n2Eim−1∏i=1n2′Fin−1xmynm!n!.
Using the identity,(44)F10A;−;t=1−t−A,and after simplifications, the right side of equation (40) is obtained. This ends the proof of equation (40). The identity equation (41) is proved in a similar manner.Theorem 11.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that AiBj=BjAi; DiEj=EjDi; DiFj=FjDi; and EiFj=FjEi, and Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0. Then, the infinite summation formulas of the (K de FHMF) hold true:(45)Fm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx+t,y=∑k=0∞AkBktkk!×Fm2;n2,n2′m1;n1,n1′;D+kI:E+kI,FA+kI:B+kI,Cx,yDk−1Ek−1.Proof:.
From the definition of the (K de FHMF) and the transformationAkA+kIm=Ak+m, the right side of equation (45) is expressed as(46)∑k,m,n=0∞∏i=1m1Aim+n+k∏i=1n1Bim+k∏i=1n1′Cin∏i=1m2Dim+n+k−1∏i=1n2Eim+k−1∏i=1n2′Fin−1xmynm!n!k!tk.
Replacem+k by l in the above result. After some simplifications, we have(47)∑l,n=0∞∏i=1m1Ail+n∏i=1n1Bil∏i=1n1′Cin∏i=1m2Dil+n−1∏i=1n2Eil−1∏i=1n2′Fin−1ynl!n!∑k=0llkxl−ktk=∑l,n=0∞∏i=1m1Ail+n∏i=1n1Bil∏i=1n1′Cin∏i=1m2Dil+n−1∏i=1n2Eil−1∏i=1n2′Fin−1x+tlynl!n!.
Using the relation(48)∑k=0llkxkyl−k=x+yl,in the inner summation, finishes the proof of equation (45).Theorem 12.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that Di+kI, Ei+kI, and Fi+kI are invertible for each integer k≥0. Then, the infinite summation formulas of the (K de FHMF) hold true:(49)∑k=0∞Bikk!−tkFm2;n2,n2′m1;n1,n1′;D:E,FA:−kI,Bi,C1+ttx,y=1+t−BiFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y,where AjBi=BiAj, i=1,…,n1.Proof:.
From the definition of the (K de FHMF), the left side of equation (49) can be expressed as(50)∑k,n=0∞∑m=0kBik−kImm!n!k!∏i=1m1Aim+n∏j=1,≠in1Bim∏i=1n1′Cin∏i=1m2Dim+n−1∏i=1n2Eim−1∏i=1n2′Fin−11+ttxmyn−tk.
Takingk=m+l, changing the summation order and simplifying, we get(51)∑m,n=0∞F10;−Bi+mI−t∏i=1m1Aim+n∏i=1n1Bim∏i=1n1′Cin∏i=1m2Dim+n−1∏i=1n2Eim−1∏i=1n2′Fin−1xmynm!n!1+tm.
Evaluating the innerF10-series in the above equation by equation (44),(52)F10Bi+mI;−;−t=1+t−Bi−mI,and simplifying, we get the right side of this theorem. This completes the proof.Theorem 13.
LetAi, Bi, Ci, Di, Ei, and Fi be the positive stable matrices in ℂr×r, such that Di+kI, Ei+kI, and Fi+kI are invertible for all integers k≥0. Then, the infinite summation formulas of the (K de FHMF) hold true:(53)∑k=0∞Bikk!t+xx−1kFm2;n2,n2′m1;n1,n1′;D:E,FA:−kI,Bi,C1+tt+xx,y=1−xt+1BiFm2;n2,n2′m1;n1,n1′;D:E,FA:B,Cx,y,where AjBi=BiAj, i=1,…,n1.Proof:.
It is similar to Theorem12. We omit the details.


## 6. Conclusion

We have investigated recursion formulas and some finite matrix and infinite matrix summation formulas involving the (K de FHMF). We remark that by specializing the sequence of matrices in the (K de FHMF), we can deduce recursion formulas and finite matrix and infinite matrix summation formulas for some Appell matrix functions [17–19].



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