# A Study on Generalized Multivariable Mittag-Leffler Function via Generalized Fractional Calculus Operators

Authors: D. L. Suthar; Mitku Andualem; Belete Debalkie
Journal: Journal of Mathematics
 (2019)

---

## Abstract

We study some properties of generalized multivariable Mittag-Leffler function. Also we establish two theorems, which give the images of this function under the generalized fractional integral operators involving Fox’s H-function as kernel. Relating affirmations in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Some known special cases have also been mentioned in the concluding section.


---

## Body

## 1. Introduction and Preliminaries

Recently, Gurjar et al. [1] introduced a multivariable generalized Mittag-Leffler (M-L) function; this function and its special cases have recently found various essential applications in solving problems in physics, biology, engineering and applied sciences (see; [2–5]). The function is defined for τ,δj,γj,lj∈C and min1≤j≤sRτ,Rδj;Rγj,Rlj>0,aj,bj>0;aj<bj+Rδj;j=1,2,…,s as(1)Eδj;τ;bjγj;lj;ajz1,…,zs=Eδ1,⋯,δs;τ;b1,⋯,bsγ1,⋯,γs;l1,⋯ls;a1,⋯asz1,…,zs=∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjz1r1⋯zsrsl1b1r1⋯lsbsrs,where(2)γar=aar∏n=1aγ+n-1ar,a∈N,r∈N0≔N∪0.Some important special cases of the multivariable generalized M-L function are enumerated below:(1)
Iflj=bj=1 in (1), the function reduces to multivariate analogue of generalized M-L function which was defined by Saxena et al. [6] as(3)Eδj;τ;1γj;1;ajz1,…,zs=Eδj;τγj;ajz1,…,zs=∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjz1r1⋯zsrsr1!⋯rs!,where τ,δj,γj∈C and Rδj>0;Raj>0;j=1,2,…,s.(2)
If we setlj=aj=bj=1 in (1), then the function reduces to another type of multivariate generalized M-L function which was also defined by Saxena et al. [6] as(4)Eδj;τ;1γj;1;1z1,…,zs=Eδj,τγjz1,…,zs=∑r1,…,rs=0∞γ1r1⋯γsrsΓτ+∑j=1sδjrjz1r1⋯zsrsr1!⋯rs!,where τ,δj,γj∈C and Rδj>0;j=1,2,…,s.(3)
If we puts=1, equation (1) reduces to generalized M-L function which was defined by Salim and Faraj [7] as(5)Eδ;τ;bγ;l;az=∑r=0∞γarΓτ+δrzrlbr,where τ,δ,γ,l∈C;min⁡Rτ,Rδ;Rγ,Rl>0;a,b>0,b≤Rδ+a.(4)
By settings=1,γ1=1,l1=b1=1, and a1=a in (1) considered by Shukla and Prajapati [8], in addition to that, if a1=1, defined by Prabhakar [9].(5)
Ifs=1 and γ1=l1=a1=b1=1 in Eq. (1), it reduces to Wiman’s function [10], moreover if τ=1, Mittag-Leffler function [11] will be the result.In the present paper, our aim is to study some fundamental properties of multivariable generalized M-L function defined in equation (1). For that, we consider two generalized fractional integral operators involving Fox’s H-function as kernel, defined by Kalla [12, 13] and further studied by Srivastava and Buschman [14]. Recently, Garg, Rao and Kalla [15] studied some fractional calculus properties of M-L type function, involving these fractional calculus operators. We use the following notations for the left-sided and right-sided generalized fractional integral operators:(6)R0+ξ1,ξfz=z-ξ1-ξ-1∫0ztξ1z-tξHP,QM,NΛUcj,αj1,pdk,βk1,Qftdt,and(7)R-ξ2,ξfz=zξ2∫z∞t-ξ2-ξ-1t-zξHP,QM,NΛVcj,αj1,pdk,βk1,Qftdt,where U and V represent the expressions t/zm1-t/zn and z/tm1-z/tn respectively, with m,n>0. Here the symbol HP,QM,N. stands for well-known Fox’s H-function, defined by means of the following Mellin-Barnes type integral [16]:(8)HP,QM,Nzcj,αj1,pdk,βk1,Q=12πi∫Lθςzςdς,where(9)θς=∏k=1MΓdk-βkς∏j=1NΓ1-cj+αjς∏k=M+1QΓ1-dk+βkς∏j=N+1PΓcj-αjς,and L is a suitable contour in C. The orders M,N,P,Q are integers, 1≤M≤Q,1≤N≤P and the parameters cj,dk∈R;αj>0,j=1,…,p,βk>0,k=1,…,Q are such that cjdk+l≠βkcj-l′-1,l,l′=0,1,…. For the conditions of analyticity of the H-function and other details, one can see [16, 17]. Throughout the present paper, we assume that these conditions are satisfied by the H-function.For our purpose, we recall the definition of generalized Wright hypergeometric functionpψq(z) (see, for details, Srivastava and Karlsson [18]), for z,ai,bj∈C and αi,βj∈R, with αi,βj≠0(i=1,…,p;j=1,…,q) defined as follows:(10)pψqz=pψqai,αi1,pbj,βj1,qz=∑k=0∞∏i=1pΓai+αikzk∏j=1qΓbj+βjkk!.The generalized Wright function was introduced by Wright [19] in the form of (10) under the condition:(11)∑j=1qβj-∑i=1pαi>-1.


## 2. Images of the Generalized Multivariable M-L Function under the Generalized Fractional Integral Operators

In this section, we consider two generalized fractional integral operators involving the Fox’s H-function as the kernels and derived the following theorems:Theorem 1.
Letτ,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRτ,Rδj,Rγj,Rlj>0,aj,bj>0;aj<bj+Rδj;j=1,2,…,s and R0+ξ1,ξ be the generalized left-side fractional integral operator (6), then there hold the result true:(12)
R
0
+
ξ
1
,
ξ
z
ε
-
1
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
v
1
,
…
,
ω
s
z
v
s
z
=
z
ε
-
1
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
v
1
,
…
,
ω
s
z
v
s
×
H
P
+
2
,
Q
+
1
M
,
N
+
2
Λ
-
ξ
1
-
ε
-
∑
i
=
1
s
v
i
r
i
,
m
,
-
ξ
,
n
,
c
j
,
α
j
1
,
p
d
k
,
β
k
1
,
Q
,
-
ξ
1
-
ξ
-
ε
-
∑
i
=
1
s
v
i
r
i
-
1
,
m
+
n
,provided that(i)
minξ,ξ1,ε,v,m,n>0(ii)
ξ1+ε+mmin1≤k≤Mdk/βk+1>0(iii)
ε+nmin1≤k≤Mdk/βk+1>0(iv)
A>0,argΛ<Aπ/2,
whereA=∑j=1N‍αj-∑j=N+1P‍αj+∑k=1M‍βk-∑k=M+1Q‍βk.Proof.
Using the definition (6) in the left hand side of (12), writing the functions in the form given by (1) and (8), interchanging the order of integration and summations under the statement of Theorem 1, we obtain(13)R0+ξ1,ξzε-1Eδj;τ;bjγj;lj;ajω1zv1,…,ωszvsz=z-ξ1-ξ-1∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjω1r1⋯ωsrsl1b1r1⋯lsbsrs12πi∫LθςΛς×∫0ztξ1+ε+v1r1+⋯+vsrs-1z-tξtzmς1-tznςdtdς,To evaluate the t-integral substituting t/z=x,dt=zdx, we obtain(14)R0+ξ1,ξzε-1Eδj;τ;bjγj;lj;ajω1zv1,…,ωszvsz=zε-1∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjω1r1⋯ωsrsl1b1r1⋯lmbsrs12πi∫LθςΛς×zv1r1+⋯+vsrs∫01xξ1+ε+v1r1+⋯+vsrs+mς-11-xξ+nςdxdς,Finally on evaluating the x integral as beta integral and re-interpreting the result in terms of H-Function and generalized multivariable M-L function, we easily arrive at the result (12).Theorem 2.
Assumeτ,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRτ,Rδj,Rγj,Rlj>0,aj,bj>0;aj<bj+Rδj;j=1,2,…,s and R-ξ2,ξ be the generalized right-side fractional integral operator (7), then(15)
R
-
ξ
2
,
ξ
z
-
ε
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
-
v
1
,
…
,
ω
s
z
-
v
s
z
=
z
-
ε
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
-
v
1
,
…
,
ω
s
z
-
v
s
×
H
P
+
2
,
Q
+
1
M
,
N
+
2
Λ
-
ξ
2
-
ε
-
∑
i
=
1
s
v
i
r
i
,
m
,
-
ξ
,
n
,
c
j
,
α
j
1
,
p
d
k
,
β
k
1
,
Q
,
-
ξ
2
-
ξ
-
ε
-
∑
i
=
1
m
v
i
r
i
,
m
+
n
,provided that(i)
minξ,ξ2,ε,v,m,n>0(ii)
ξ2+ε+mmin1≤k≤Mdk/βk+1>0(iii)
ε+nmin1≤k≤Mdk/βk+1>0(iv)
A>0,argΛ<Aπ/2
whereA=∑j=1N‍αj-∑j=N+1P‍αj+∑k=1M‍βk-∑k=M+1Q‍βk.Proof.
Proceeding as in Theorem1, one can easily prove the Theorem 2. Therefore, we omit the detailed proof of Theorem 2.


## 3. Images of the Generalized Multivariable M-L Function under the Saigo’s Fractional Integral Operators

If we chooseM=1,N=P=Q=2,c1=1-ϑ-κ,c2=1+η,d1=0,d2=1-ϑ,α1=α2=β1=β2=1,m=0,n=1,τ=-1 and v=1 in the Theorems 1 and 2, then the fractional integral operators R0+ξ1,ξ and R-ξ2,ξ reduce to the corresponding Saigo’s operators [20, 21]. These operators are connected by the following functional relations:(16)R0+0,ϑ-1fz=Γϑ+κΓ-ηI0+ϑ,κ,ηfz,and(17)R-0,ϑ-1fz=Γϑ+κΓ-ηI-ϑ,κ,ηfz,where the operators I0+ϑ,κ,ηf and I-ϑ,κ,ηf respectively denote the Saigo’s left-side and right-side fractional integral operators and are defined as(18)I0+ϑ,κ,ηfz=z-ϑ-κΓϑ∫0zz-tϑ-12F1ϑ+κ,-η;ϑ;1-tzftdt,ϑ,κ,η∈C,Rϑ>0 and(19)I-ϑ,κ,ηfz=1Γϑ∫z∞t-zϑ-1t-ϑ-κ2F1ϑ+κ,-η;ϑ;1-ztftdt,ϑ,κ,η∈C,Rϑ>0.Lemma 3.
Letϑ,κ,η∈C. Then their exists the relation (1)
IfRϑ>0 and Rε>max0,Rκ-η, then(20)I0+ϑ,κ,ηtε-1z=ΓεΓε+η-κΓε-κΓε+ϑ+ηzε-κ-1.(2)
IfRϑ>0 and Rε<1+minRκ,Rη, then(21)I-ϑ,κ,ηtε-1z=Γ1-ε+κΓ1-ε+ηΓ1-εΓ1-ε+ϑ+κ+ηzε-κ-1.Now, on using the above mentioned substitutions and relations, we obtain the following images of the generalized multivariable M-L function under the fractional integral operators of Saigo type:Corollary 4.
Letτ,ϑ,κ,η,ε,δj,γj,lj,ωj∈C; such that min1≤j≤sRϑ,Rτ,Rδj,Rγj,Rlj>0,Rε>max0,Rκ-η;aj,bj>0;aj<bj+Rδj;j=1,2,…,s, then the following formula holds:(22)
I
0
+
ϑ
,
κ
,
η
z
ε
-
1
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
v
1
,
…
,
ω
s
z
v
s
z
=
z
ε
-
κ
-
1
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
2
s
+
2
ψ
s
+
3
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
,
∑
j
=
1
s
v
j
,
ε
+
η
-
κ
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
-
κ
,
∑
j
=
1
s
v
j
,
ε
+
ϑ
+
η
,
∑
j
=
1
s
v
j
ω
1
z
v
1
⋮
ω
s
z
v
s
.Proof.
Denote L.H.S. of Corollary4 by l1. By virtue of (1) and (18), we have(23)l1=I0+ϑ,κ,ηzε-1∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjω1r1zv1r1⋯ωsrszvsrsl1b1r1⋯lsbsrsz=∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjω1r1⋯ωsrsl1b1r1⋯lsbsrsI0+ϑ,κ,ηzε+v1r1+⋯+vsrs-1z,which upon Lemma 3(1), yields(24)l1=zε-κ-1∑r1,…,rs=0∞γ1a1r1⋯γsasrsΓτ+∑j=1sδjrjω1r1⋯ωsrsl1b1r1⋯lsbsrs×Γε+v1r1+⋯+vsrsΓε+v1r1+⋯+vsrs+η-κΓε+v1r1+⋯+vsrs-κΓε+v1r1+⋯+vsrs+ϑ+ηzv1r1+⋯+vsrs,Using the definition of (10) in the right-hand side of (24), we arrive at the result (22).Corollary 5.
Assumeτ,ϑ,κ,η,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRϑ,Rτ,Rδj,Rγj,Rlj>0,Rε>max-Rκ,-Rηaj,bj>0;aj<bj+Rδj;j=1,2,…,s, then(25)
I
-
ϑ
,
κ
,
η
z
-
ε
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
-
v
1
,
…
,
ω
s
z
-
v
s
z
=
z
-
ε
-
κ
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
2
s
+
2
ψ
s
+
3
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
+
κ
,
∑
j
=
1
s
v
j
,
ε
+
η
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
,
∑
j
=
1
s
v
j
,
ε
+
ϑ
+
κ
+
η
,
∑
j
=
1
s
v
j
ω
1
z
-
v
1
⋮
ω
s
z
-
v
s
.Proof.
By a similar manner as in proof of Corollary4 by using Lemma 3(2), we get the desired formula (25).


## 4. Images of the Generalized Multivariable Mittage-Leffler Function under the Erdélyi-Kober Fractional Integral Operators

If we takeκ=0 in the Saigo’s fractional integral operators (18) and (19), then due to Saigo [20], we get(26)I0+ϑ,0,ηfz=I+ϑ,ηfzand(27)I-ϑ,0,ηfz=I-ϑ,ηfz,where the Erdélyi-Kober fractional integral operators are defined by(28)I+ϑ,ηfz=z-ϑ-ηΓϑ∫0zz-tϑ-1tηftdtRϑ>0,and(29)I-ϑ,ηfz=zηΓϑ∫z∞t-zϑ-1t-ϑ-ηftdtRϑ>0.We now give images of the generalized multivariable M-L function under the Erdélyi-Kober fractional integral operators:Corollary 6.
Letτ,ϑ,η,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRϑ,Rτ,Rδj;Rγj,Rlj>0,Rε>R-η,aj,bj>0;aj<bj+Rδj;j=1,2,…,s, then(30)
I
0
+
ϑ
,
η
z
ε
-
1
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
v
1
,
…
,
ω
s
z
v
s
z
=
z
ε
-
1
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
×
2
s
+
1
ψ
s
+
2
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
+
η
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
+
ϑ
+
η
,
∑
j
=
1
s
v
j
ω
1
z
v
1
⋮
ω
s
z
v
s
.Corollary 7.
Letτ,ϑ,η,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRϑ,Rτ,Rδj;Rγj,Rlj>0,Rε>-Rη,aj,bj>0;aj<bj+Rδj;j=1,2,…,s, then(31)
I
-
ϑ
,
η
z
-
ε
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
-
v
1
,
…
,
ω
s
z
-
v
s
z
=
z
-
ε
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
×
2
s
+
1
ψ
s
+
2
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
+
η
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
+
ϑ
+
η
,
∑
j
=
1
s
v
j
ω
1
z
-
v
1
⋮
ω
s
z
-
v
s
.


## 5. Special Cases

In this section, we consider some consequences and applications of the results derived in the previous sections. If we takeϑ+κ=0, then the fractional calculus operators (18) and (19), respectively, reduce to the Riemann-Liouville and Weyl fractional integral operators. Hence, we obtain the following image formulas:Corollary 8.
Letτ,ϑ,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRϑ,Rτ,Rδj;Rγj,Rlj>0,aj,bj>0,aj<bj+Rδj;j=1,2,…,s, then the following formula holds:(32)
I
ϑ
z
ε
-
1
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
v
1
,
…
,
ω
s
z
v
s
z
=
z
ε
+
ϑ
-
1
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
×
2
s
+
1
ψ
s
+
2
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
+
ϑ
,
∑
j
=
1
s
v
j
ω
1
z
v
1
⋮
ω
s
z
v
s
,where the Riemann-Liouville fractional integral operator is defined by(33)Iϑfz=I0+ϑ,-ϑ,ηfz=1Γϑ∫0zz-tϑ-1ftdt.Corollary 9.
Assumeτ,ϑ,ε,δj,γj,lj,ωj∈C such that min1≤j≤sRϑ,Rτ,Rδj,Rγj,Rlj>0,aj,bj>0;aj<bj+Rδj;j=1,2,…,s, then the following result is true:(34)
K
ϑ
z
-
ε
E
δ
j
;
τ
;
b
j
γ
j
;
l
j
;
a
j
ω
1
z
-
v
1
,
…
,
ω
s
z
-
v
s
z
=
z
-
ε
+
ϑ
Γ
l
1
⋯
Γ
l
s
Γ
γ
1
⋯
Γ
γ
s
×
2
s
+
1
ψ
s
+
2
γ
1
,
a
1
,
⋯
,
γ
s
,
a
s
,
ε
-
ϑ
,
∑
j
=
1
s
v
j
,
1,1
,
⋯
,
1,1
︷
s
t
i
m
e
s
l
1
,
b
1
,
⋯
,
l
s
,
b
s
,
τ
,
∑
j
=
1
s
δ
j
,
ε
,
∑
j
=
1
s
v
j
ω
1
z
-
v
1
⋮
ω
s
z
-
v
s
,where the Weyl type fractional integral operator is given by(35)Kϑfz=I-ϑ,-ϑ,ηfz=1Γϑ∫z∞t-zϑ-1ftdt.


## 6. Concluding Remark

The generalized multivariable M-L function is interesting due to the various (described in introduction section) M-L functions that follow as its particular cases, so the generalized fractional calculus formulas are deduced in this communication, and we can find many applications giving the Saigo, Erdelyi-Kober, Riemann-Liouville and Weyl type fractional integrals of aforementioned functions on taking special cases into account. For various other special cases we refer to [22–27] and we left results for the interested readers.



---

*Source: 9864737-2019-08-01.xml*