# On Multi-Index Mittag–Leffler Function of Several Variables and Fractional Differential Equations

Authors: B. B. Jaimini; Manju Sharma; D. L. Suthar; S. D. Purohit
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we have studied a unified multi-index Mittag–Leffler function of several variables. An integral operator involving this Mittag–Leffler function is defined, and then, certain properties of the operator are established. The fractional differential equations involving the multi-index Mittag–Leffler function of several variables are also solved. Our results are very general, and these unify many known results. Some of the results are concluded at the end of the paper as special cases of our primary results.


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## Body

## 1. Introduction

Recently, Mittag–Leffler (M-L) functions have demonstrated their special connection to fractional calculus, with a particular emphasis on fractional calculus problems arising from implementations. Several new special functions and implementations have been discovered over the last few decades. The advancement of research in the new era of special functions and their applications in mathematical modelling continues to attract many scientists from various disciplines (see recent papers; [1–13]).The Mittag–Leffler function is extended to multi-index function in the following form [14, 15]:(1)Eγ,Kα1,β1,…,αm,βm;z=∑n=0∞γKnzn∏j=1mΓβj+nαjn!,where αj,βj,γ∈ℂ;ℜαj>0;ℜβj>0j=1,…,m;ℜ∑j=1mαj>0; and K is an arbitrary complex number, i.e., K∈ℂ.If we makeγ=K=1 in (1) it reduces to the multi-index M-L function studied by Kiryakova [16, 17].A multivariable extension of Mittag–Leffler function widely studied by Gautam [18], and also by Saxena et al. ([19], p. 547, Equation (7.1)), is defined and represented as follows:(2)Eρr,λγr,lrz1,…,zr=∑k1,…kr=0∞γ1l1k1…γrlrkrΓλ+k1ρ1+⋯+krρrz1k1…zrkrk1!…kr!,where λ,γj,lj,ρj∈ℂℜρj>0;ℜlj>0;λ∉ℤ0−=0,−1,−2,…; and j=1,2,…,r.Motivated by the work on these functions, we consider here the subsequent multivariable and multi-index Mittag–Leffler function:(3)Eρ1r,…,ρmr;β1,…,βmγr,lrz1,…,zr=∑k1,…,kr=0∞∏i=1rγiliki∏j=1mΓβj+∑i=1rρjikiz1k1k1!…zrkrkr!,where βj,γi,li,ρji∈ℂ; ℜρji>0;ℜli>0;βj∉ℤ0−=0,−1,−2,…; ρjr≡ρj′,ρj″,…,ρjr;i=1,…,r;andj=1,…,m.We have also studied here, the integral operator involving the function defined by (3), as follows:(4)Eρ1r,…,ρmr;β1,…,βm;ωr;a+γr,lrψx=∫axx−tβ1−1Eρ1r,…,ρmr;β1,…,βmγr,lrω1x−tρ1′,…,ωrx−tρ1rψtdt,with ωi,ρji,γi,li,βj∈ℂ;x>a;ωix−tρ1i<1;ℜρji>0;j=1,…,m;andi=1,…,r.The Riemann–Liouville fractional derivative operatorD0+α is defined as follows [20]:(5)Da+αψx=ddxnIa+n−αψx,α∈ℂ;ℜα>0;n=ℜα+1,where Ia+αψx is the fractional integral operator defined by(6)Ia+αψx=1Γα∫axψtx−t1−αdt.The elementary definitions are also required to be mentioned as follows.The Laplace transform of fractional derivativeD0+αfx is given as(7)LD0+αf;s=sαFs−∑k=1nsk−1D0+α−kf0+,ℜs>0;n−1<α<n.Also, the formula for Laplace transform is(8)dndsnLyx:s=−1nLxnyxs.


## 2. Results Required

The integral for the generalized M-L function defined in (3) is given by(9)1Γσ∫0xx−tβ1−1tσ−1Eρ1r,…,ρmr;β1,…,βmγr,lrω1x−tρ1′,….,ωrx−tρ1rdt=xβ1+σ−1Eρ1r,…,ρmr;β1+σ,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r.The result in (9) is established in view of definition in (3) and using the elementary beta integral.The Laplace transform ofEρ1r,…,ρmr;β1,…,βmγr,lrω1xρ1′,…,ωrxρ1r defined in (3), easily obtained here, is as follows:(10)Lxβ1−1Eρ1r,…,ρmr;β1,…,βmγr,lrω1xρ1′,…,ωrxρ1rx=s−β1∏j=2m1Γβj×Fm0−1::0;…;01;…;1−:γ1,l1;…;γr,lr;βj;ρj′,…,ρjr2,m:−;…;−;ω1s−ρ1′,…,ωrs−ρ1r,where α,βj,ρji,γi,li,ωi∈ℂ;ℜs>0;ℜρji>0;ℜβj;ℜli>0;j=1,…,m;i=1,…,r; and βj:ρj′,…,ρjr2,m≡β2;ρ2′,…,ρ2r,…,βm;ρm′,…,ρmr.Here,Fω1s−ρ1′,…,ωrs−ρ1r is the generalized Lauricella function ([21], p. 37, Equations (21–23)).


## 3. Main Results

Theorem 1.
Leta∈ℝ+;α,βj,ρji,γi,li,ωi∈ℂ;ℜα>0;ℜρji>0;ℜβj>0;andℜli>0j=1,…,m;i=1,…,r.
Then, forx>a, we have(11)Da+αt−aβ1−1Eρ1r,…,ρmr;β1,…,βmγr,lrω1t−aρ1′,…,ωrt−aρ1rx=x−aβ1−α−1Eρ1r,…,ρmr;β1−α,β2,…,βmγr,lrω1x−aρ1′,…,ωrx−aρ1rand(12)Ia+αt−aβ1−1Eρ1r,…,ρmr;β1,…βmγr,lrω1t−aρ1′,…,ωrt−aρ1rx=x−aβ1+α−1Eρ1r,…,ρmr;β1+α,β2,…,βmγr,lrω1x−aρ1′,…,ωrx−aρ1r.
Ifα,βj,γi,li,ωi∈ℂ;ℜα>0;ℜρji>0;ℜβj>0;ℜli>0;j=1,…,m;andi=1,…,r with the initial condition I0+1−αy0+=c (c is an arbitrary constant) and solution of differential equations existing in the space L0,∞, then Theorems 2–4 are stated in the following form.Theorem 2.
If(13)D0+αyx=λEρ1r,…,ρmr;β1,…,βm;ωr;0+γr,lrx+fx,then its solution is given by(14)yx=cxα−1Γα+λxβ1+αEρ1r,…,ρmr;β1+α+1,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r+1Γα∫0xx−tα−1ftdt.Theorem 3.
If(15)D0+αyx=λEρ1r,…,ρmr;β1,…,βm;ωr;0+γr,lrx+pxβ1Eρ1r,…,ρmr;β1+1,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r,then its solution is given by(16)yx=cxα−1Γα+λ+pxβ1+αEρ1r,…,ρmr;β1+α+1,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r.Theorem 4.
If(17)xD0+αyx=λEρ1r,…,ρmr;β1,…,βm;ωr;0+γr,lrx,then its solution is given by(18)yx=cxα−1Γα+λΓα∫0xtα−1x−tβ1−1×Eρ1r,…,ρmr;β1+1,β2,…,βmγr,lrω1x−tρ1′,…,ωrx−tρ1rdt.Proof.
In Theorem1, let the left-hand side of result (11) be Δ1, i.e.,(19)Δ1=Da+αt−aβ1−1Eρ1r,…,ρmr;β1,…,βmγr,lrω1t−aρ1′,…,ωrt−aρ1rx.
Having used the definition ofEρ1r,…,ρmr;β1,…,βmγr,lr. given in (3), we obtain the following form:(20)Δ1=∑k1,...,kr=0∞∏i=1rγilikiωiki∏j=1mΓβj+∑i=1rρjiki∏i=1rki!Da+αt−aβ1+∑i=1rρ1iki−1x.
On using the fractional derivative of power functiont−aβ1+∑i=1rρ1iki−1 ([20], p. 36, Equation (2.26)), we have(21)Δ1=x−aβ1−α−1∑k1,...,kr=0∞∏i=1rγilikiωikix−a∑i=1rρ1iki∏j=2mΓβj+∑i=1rρ1ikiΓβ1−α+∑i=1rρ1iki∏i=1rki!.
On interpreting multiple series by the definition ofEρ1r,…,ρmr;β1,…,βmγr,lr⋅, we at once arrive at (11).
The proof of (12) follows the proof of (11) using (6) and ([20], p. 40, Equation (2.44)) therein.
Theorem2 is proved as follows.
Use the definition of operatorEρ1r,…,ρmr;β1,…,βm;ωr;a+γr,lrψx (at a=0 and ψx=1) and result (9) (at σ=1) in (13), we have(22)D0+αyx=λxβ1Eρ1r,…,ρmr;β1+1,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r+fx.
Taking Laplace transform of (22) and then using formula (7) (for n = 1) and (10), therein we have(23)ys=cs−α+λs−β1−1−α∏j=2m1Γβj×Fm−1:0;…;00:1;…;1−:γ1,l1;…;γr,lr;βj;ρj′,…,ρjr2,m:−;…;−;ω1s−ρ1′,…,ωrs−ρ1r+fss−α.
In view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we have the form(24)ys=cs−α+λ∑k1,...,kr=0∞∏i=1rγilikiωikis−β1−1−α−∑i=1rρjiki∏j=2mΓβj+∑i=1rρjiki∏i=1rki!+fss−α.
Applying inverse Laplace transform on both sides of (24) and using convolution theorem, we find(25)yx=cxα−1Γα+λ∑k1,...,kr=0∞∏i=1rγilikiωiki∏j=1mΓβj+∑i=1rρjiki∏i=1rki!xβ1+α+∑i=1rρjikiΓβ1+1+α+∑i=1rρjiki+1Γα∫0xftx−t1−αdt.
Now, on interpreting the multiple series using (3), we at once arrive at desired result (14).Proof.
of Theorem3. We use (at a=0 and ψx=1) and (9) (at σ=1) in (15), and it takes the following form:(26)D0+αyx=λ+pxβ1Eρ1r,…,ρmr;β1+1,β2,…,βmγr,lrω1xρ1′,….,ωrxρ1r.
On both sides of (26), we take Laplace transform and then using formula (7) (for n = 1) and (10) therein, we obtain(27)ys=cs−α+λ+ps−β1−1−α=∏j=2m1Γβj×Fm−1:0;…;00:1;…;1−:γ1,l1;…;γr,lr;βj;ρj′,…,ρjr2,m:−;…;−;ω1s−ρ1′,…,ωrs−ρ1r.
In view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we have(28)ys=cs−α+λ+p∑k1,...,kr=0∞∏i=1rγilikiωikis−β1−1−α−∑i=1rρjiki∏j=1mΓβj+∑i=1rρjiki∏i=1rki!.
Applying inverse Laplace transform on (28), we have(29)yx=cxα−1Γα+λ+p∑k1,...,kr=0∞∏i=1rγilikiωiki∏j=2mΓβj+∑i=1rρjiki∏i=1rki!×xβ1+α+∑i=1rρjikiΓβ1+1+α+∑i=1rρjiki.
On interpreting the multiple series using (3), we at once arrive at result (16).Proof.
of Theorem4. We use operator Eρ1r,…,ρmr;β1,…,βm;ωr;a+γr,lrψx (at a=0 and ψx=1) and (9) (at σ=1) in (17), and we have the following form:(30)xD0+αyx=λxβ1Eρ1r,…,ρmr;β1+1,β2,…,βmγr,lrω1xρ1′,…,ωrxρ1r.
On both sides of (30), we take Laplace transform and use formulae (8) and (10) (for n = 1); then, we obtain(31)ddsys+αsys=−λs−β1−1−α∏j=2m1Γβj×Fm−1:0;…;00:1;…;1−:γ1,l1;…;γr,lr;βj;ρj′,…,ρjr2,m:−;…;−;ω1s−ρ1′,…,ωrs−ρ1r.
In view of the definition of generalized Lauricella function ([21], p. 37, Equations (21–23)), we have(32)ddsys+αsys=−λ∑k1,...,kr=0∞∏i=1rγilikiωiki∏j=2mΓβj+∑i=1rρjiki∏i=1rki!s−β1−1−α−∑i=1rρjiki.
Since this is a linear differential equation of first order and first degree,(33)ys=λ∑k1,...,kr=0∞∏i=1rγilikiωiki∏j=2mΓβj+∑i=1rρjiki∏i=1rki!s−β1−α−∑i=1rk1ρ1iβ1+∑i=1rkiρ1i+cs−α.
Taking inverse Laplace transform of (33), we have(34)yx=λ∑k1,...kr=0∞∏i=1rγilikiωiki∏j=2mΓβj+∑i=1rρjiki∏i=1rki!1/β1+∑i=1rkiρ1i×L−1s−α−β1−∑i=1rk1ρ1i+cL−1s−α.
In view of convolution theorem, we obtain(35)yx=λΓα∑k1,...kr=0∞∏i=1rγilikiωiki∏j=2mΓβj+∑i=1rkiρji∏i=1rki!1/Γβ1+1+∑i=1rkiρji.×∫0xtα−1x−tβ1−1+∑i=1rkiρ1idt+cΓαxα−1.
Now, on interpreting the multiple series in view of (3), we obtain the result in (18).


## 4. Conclusion

Here, we conclude further interesting known results:(1)
Our main results form=1, respectively, give the known results provided by Gupta and Jaimini ([22], pp. 145–146, Equations (1–10)).(2)
If in result (10) and in Theorem 2, we take r=1i.e.,ω2=,…,=ωr=0, then result (10) reduces to the known result provided by Saxena et al. ([23], p. 10, Equation (50)) and Theorem 2 gives the correct form (at ν=0) of the theorem provided by Saxena et al. ([23], p. 10, Theorem (5.1)).(3)
Form=1 and r=1, Theorems 1 to 4 reduce, respectively, to the known results (at ν=0) provided by Srivastava and Tomovski [24].(4)
If in Theorem1, we take m=1 and l1=l2=,…,=lr=1, then these results, respectively, reduce to the known results provided by Gautam ([18], pp. 201–202, Equations (4.64)–(4.65)).(5)
If in Theorems1 to 4, we take m=1, l1=l2=,…,=lr=1, and ρ1′=ρ1′′=,…,=ρ1r=1, then these, respectively, reduce to the results for function ϕ2r⋅ provided by Gupta ([25], pp. 250–253, Equations (4.9.19)–(4.9.27)).Therefore, the results presented in the article would immediately yield a large number of results that include a wide range of special functions occurring in issues of scientific research, computer science, and applied mathematics, among others.



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