R E S E A R C H
Open Access
Generalized fractional integral operators
and the multivariable
H
-function
Praveen Agarwal
1, Sergei V Rogosin
2, Erkinjon T Karimov
3*and Mehar Chand
4*Correspondence:
3Institute of Mathematics, National
University of Uzbekistan, Tashkent, 100125, Uzbekistan
Full list of author information is available at the end of the article
Abstract
The main object of the present paper is to establish new fractional integral formulas (of Marichev-Saigo-Maeda type) involving the products of the multivariable
H-functions and the first class of multivariable polynomials due to Srivastava and Garg. All the results derived here are of general character and can yield a number of (new and known) results in the theory of fractional calculus.
MSC: 26A33; 33C45; 33C60; 33C70; 33E12
Keywords: Marichev-Saigo-Maeda fractional integral operator; multivariable
H-function; first class of multivariable polynomials; Mittag-Leffler function
1 Introduction
Nowadays, fractional calculus is used as a mathematical tool for modeling the processes in engineering, mathematical economics, physics, chemistry, biology and other branches of science (see [–]). Motivated by the above mentioned works, here we aim to es-tablish two new fractional integral formulas (of Marichev-Saigo-Maeda type) involving the product of multivariableH-functions and the first class of multivariable polynomials Sm,...,ms
n (x, . . . ,xs). The so-called Marichev-Saigo-Maeda generalized fractional operators
were introduced nearly years ago by Marichev [] and studied in some recent pa-pers, including the papers by Saigo and Maeda [] and by Saxena and Saigo []. Due to the usefulness and the importance of the Marichev-Saigo-Maeda fractional integral operators, many authors have presented a number of interesting integral formulas involv-ing special functions by usinvolv-ing the Marichev-Saigo-Maeda fractional integral operator (see [, –]). By virtue of the unified nature of Marichev-Saigo-Maeda fractional integral operators, a large number of new and known results involving Saigo, Riemann-Liouville and Erdélyi-Kober fractional integral operators follow as special cases of our main formu-las.
Our main results consist in application of the Marichev-Saigo-Maeda generalized frac-tional operators:
I,xα,α,β,β,ηf(x)
= x
–α
(η)
x
(x–t)η–t–αF
α,α,β,β;η; – t x, –
x t
f(t)dt (Reη> ) (.)
and
Ix,∞α,α,β,β,ηf(x)
= x
–α
(η)
∞ x
(t–x)η–t–αF
α,α,β,β;η; –x t, –
t x
f(t)dt (Reη> ), (.)
to the multivariable H-functions (see,e.g., [, ]) and the first class of multivariable polynomialsSm,...,ms
n (x, . . . ,xs) given by (.).
A variant of such operators (integral transforms) was introduced by Marichev [] as Mellin type convolution operators with a special functionF(·) in the kernel. These
oper-ators were rediscovered and studied by Saigo in [] as a generalization of the so-called Saigo fractional integral operators, see []. The properties of these operators were studied by Saigo and Maeda []. In particular, relations of the operators with the Mellin trans-forms, hypergeometric operators (or Saigo fractional integral operators), their decompo-sitions and acting properties in the McBride spacesFp;μ(see []) were found.
In (.), (.) the symbolF(·) denotes the so-called rd Appell function (known also as
Horn function) (see [], p.):
F
α,α;β,β;η;x;y=
∞
m,n=
(α)m(α)n(β)m(β)n
(η)m+nm!n!
xmyn max|x|,|y| < . (.)
The properties of this function are discussed in [], pp.-. In particular, its relation to the Gauss hypergeometric function is presented:
F(α,η–α;β,η–β;η;x;y) =F(α,β;η;x+y–xy).
It is known that the rd Appell function cannot be expressed as a product of twoF
func-tions and satisfy pairs of linear partial differential equafunc-tions of the second order.
Operators (.), (.) can be reduced to the fractional integral operators introduced by Saigo [] due to the following relations:
I,xα,,β,β,γf(x) = I,xγ,α–γ,–βf(x) (γ ∈C) (.)
and
Ix,∞α,,β,β,γf(x) = Ix,∞γ,α–γ,–βf(x) (γ ∈C). (.)
The multivariableH-function, introduced by Srivastava and Panda in a series of research papers (see [], p., Eq. (.), [], p., Eq. (.) and [, ]), can be defined and represented in the following manner (see,e.g., [], p., Eq. (C.)):a
H[x, . . . ,xr] =Hp,q:{p,n:{mr,qr,nr}r}
⎡ ⎢ ⎢ ⎣
x
.. . xr
(aj;αj(), . . . ,α (r)
j ),p:{(c(r)j ,γ (r) j ),pr}
(bj;βj(), . . . ,β (r)
j ),q:{(d(r)j ,δ (r) j ),qr}
⎤ ⎥ ⎥ ⎦
=
(π ω)r
L · · ·
Lr
φ(ξ, . . . ,ξr) r
i=
θi(ξi)xξiidξi
where
φ(ξ, . . . ,ξr) =
n
j=( –aj+
r i=α
(i) j ξi)
p
j=n+(aj–
r i=α
(i) j ξi)
q
j=( –bj+
r i=β
(i) j ξi)
, (.)
θi(ξi) =
ni
j=( –c (i) j +γ
(i) j ξi)
mi
j=(d (i) j –δ
(i) j ξi)
pi
j=ni+(c
(i) j –γ
(i) j ξi)
qi
j=mi+( –d
(i) j +δ
(i) j ξi)
, ∀i∈ {, , . . . ,r}. (.)
In these formulas, for anyi= , . . . ,r,Lirepresents the contours which start at the point τi–ω∞and goτi+ω∞withτi∈R= (–∞,∞) such that all the poles of(d(i)j –δ
(i) j ξi),
j= , . . . ,mi;i= , . . . ,r, are separated from those of( –c(i)j +γ (i)
j ξi),j= , . . . ,ni;i= , . . . ,r,
and( –aj+ri=α (i)
j ξi),j= , . . . ,n. Here, the integersn,p,q,mi,ni,pi, qisatisfy the
inequalities ≤n≤p;q≥, ≤mi≤qiand ≤ni≤pi,i= , . . . ,r.
Further, suppose that the parametersaj,j= , . . . ,p;c(i)j , j= , . . . ,pi;i= , . . . ,r, bj,j=
, . . . ,q;d(i)j ,j= , . . . ,qi;i= , . . . ,r, are complex numbers, while the associated coefficients α(i)j ,j= , . . . ,p;i= , . . . ,r;γj(i),j= , . . . ,pi;i= , . . . ,r,βj(i),j= , . . . ,q;i= , . . . ,r;δ
(i) j ,j=
, . . . ,qi;i= , . . . ,r, are positive real numbers such that
i= p
j= αj(i)+
pi
j= γj(i)–
q
j= βj(i)–
qi
j=
δj(i)≤, (.)
i= n
j= αj(i)–
p
j=n+ αj(i)+
q
j= βj(i)+
ni
j= γj(i)–
pi
j=ni+
γj(i)+
mi
j= δj(i)–
qi
j=mi+
δ(i)j > . (.)
It is assumed that the poles of integrand of (.) are simple. Under the conditions (.) the integrals in (.) converge absolutely (see [], p.) in the domain
arg(zi)< π
i, i= , . . . ,r, (.)
and pointszi= ,i= , . . . ,r, and various exceptional parameter values being tacitly
ex-cluded.
From Srivastava and Panda [], we have
H[z, . . . ,zr] =o
|z|e, . . . ,|zr|er, max
≤j≤r
|zj|→, (.)
whereei=min≤j≤mi[
(d(i)
j )
δj(i) ],i= , . . . ,r.
Ifn= in (.), then the following asymptotic expansion (Srivastava and Panda []) holds:
H[z, . . . ,zr] =o
|z|g, . . . ,|zr|gr, min
≤j≤r
|zj|→ ∞, (.)
wheregi=max≤j≤mi[
(c(i)
j )
γj(i) ],i= , . . . ,r, provided that (.), (.) and (.) hold true.
Remark When n= , the multivariable H-function defined by (.) reduces to the
variables is also defined and studied by Munot and Kalla [], Verma [] and Hai and Yakubovich [].
Remark It is interesting to observe that forn=p=q= , the multivariableH-function breaks up into the product ofr,H-functions; and consequently, there holds the following result (Saxena []):
H,:{mr,nr}
,:{pr,qr}
⎡ ⎢ ⎢ ⎣
x
.. . xr
– :{(c(r)j ,γj(r)),pr}
– :{(d(r)j ,δ(r)j ),qr}
⎤ ⎥ ⎥ ⎦=
r
i=
Hmi,ni
pi,qi
zi{
(c(i)j ,γj(i)),pi}
{(d(i)j ,δj(i)),qi}
. (.)
Remark The function defined by (.) was introduced and studied by Srivastava and Panda []. Whenαj()=· · ·=α(r)j ,j= , . . . ,p;βj()=· · ·=βj(r),j= , . . . ,q, in (.) the mul-tivariableH-function defined and studied by Saxena [, ] is obtained. In case all the Greek letters are assumed to be unity, theH-function of several complex variables (.) re-duces to the G-function of several complex variables studied by Khadia and Goyal [, ].
Remark Fractional integrals involving multivariableH-functions are given in a series of papers by Saigo and Saxena [–], Srivastava and Hussain [], Saigoet al.[] and others.
In the sequel, Srivastava and Garg [], p., Eq. (.), introduced the
multivari-able analogue of the polynomials Sm
n(x). This first class of multivariable polynomials
Sm,...,ms
n (x, . . . ,xs) is defined by
Sm,...,ms
n (x, . . . ,xs) =
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks(n;k, . . . ,ks)
xk · · ·xks
s
k!· · ·ks!
, (.)
wherem, . . . ,msare arbitrary positive integers and the coefficients(n;k, . . . ,ks) (n,ki≥
,i= , . . . ,s) are arbitrary constants, real or complex.
2 Main results
In this section, we prove two theorems on composition of the Marichev-Saigo-Maeda op-erators with the product of a multivariableH-function and the first class of multivariable polynomials. We start with presenting image formulas involving Marichev-Saigo-Maeda fractional integral operators.
Lemma (see []) Letα,α,β,β,γ ∈C.
If (γ) > , (p) >max[, (α+α+β–γ), (α–β)],then
I,xα,α,β,β,γxρ–=xρ–α–α+γ– (ρ)(ρ+γ–α–α
–β)(ρ+β–α)
(ρ+γ –α–α)(ρ+γ –α–β)(ρ+β). (.)
If (γ) > , (p) < +min[ (–β), (α+α–γ), (α+β–γ)],then
Iα,α,β,β,γ
x,∞ xρ–=xρ–α–α +γ–
×( +α+α–γ–ρ)( +α+β–γ–ρ)( –β–ρ)
Theorem Let (γ) > andα,α,β,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . . ,r};
j∈ {, . . . ,s}).Let the following conditions be satisfied,too: (i) |argzi|<π
i,i> (i= , . . . ,r),where
i= n
j= αj(i)–
p
j=n+ αj(i)–
q
j= βj(i)+
ni
j= γj(i)
–
pi
j=ni+
γj(i)+
mi
j= δj(i)–
qi
j=mi+
δj(i)> . (.)
(ii) min≤j≤mi,≤i≤r[–σi (d
(i)
j )
δj(i) ] < (μ) +min[ (α
), (β), (γ–α–β)].
(iii) |a
bx|< ,also we have
(μ) +
r
i=
σi min
≤j≤mi
(d(i) j ) δj(i)
>max, α+α+β–γ, α–β (i= , . . . ,r),
(η) +
r
i=
vi min ≤j≤mi
(d(i) j ) δj(i)
>max, α+α+β–γ, α–β (i= , . . . ,r).
(.)
Then the following result holds:
Iα+,α,β,β,γtμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×Hztσ(b–at)–v, . . . ,zrtσr(b–at)–vr (x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s
j=δjkjxsj=λjkjH,n+:{mr,nr};,
p+,q+:{pr,qr};,
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
zx
σ bv
.. . zrx
σr
bvr
–abx
(aj;α()j , . . . ,α (r)
j ),p,A:{(c(r)j ,γ (r) j ),pr}; –
(bj;βj(), . . . ,β (r)
j ),q,B:{(d(r)j ,δ (r)
j ),qr}; (, )
⎤ ⎥ ⎥ ⎥ ⎥
⎦, (.)
where
A=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–γ +α+α+β–
s
j=
λjkj;σ, . . . ,σr,
,
–μ+α–β–
s
j=
λjkj;σ, . . . ,σr,
and
B=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–γ +α+α–
s
j=
λjkj;σ, . . . ,σr,
–μ–γ +α+β–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–β–
s
j=
λjkj;σ, . . . ,σr,
.
Theorem Letα,α,β,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . . ,r};j∈ {, . . . ,s})
and (γ) > .Let the following conditions be satisfied,too:
(μ) –
r
i=
σi min
≤j≤mi
(d(i) j ) δ(i)j
< +max (–β), α+α–γ, α+β–γ,
(η) –
r
i=
vi≤j≤mmin
i
(d(i) j ) δj(i)
< +max (–β), α+α–γ, α+β–γ,
and the conditions(i)-(iii)in Theoremare also satisfied. Then the following result holds:
Iα–,α,β,β,γtμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×Hztσ(b–at)–v, . . . ,zrtσr(b–at)–vr (x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s j=δjkj
x
s j=λjkj
H,n+:{mr,nr};,
p+,q+:{pr,qr};,
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
zx
σ bv
.. . zrx
σr
bvr
–abx
(aj;αj(), . . . ,α (r)
j ),p,C:{(c(r)j ,γ (r) j ),pr}; –
(bj;βj(), . . . ,β (r)
j ),q,D:{(d(r)j ,δ (r)
j ),qr}; (, )
⎤ ⎥ ⎥ ⎥ ⎥
⎦, (.)
where
C=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–
s
j=
λjkj;σ, . . . ,σr,
,
+α+α+β–μ–γ –
s
j=
λjkj;σ, . . . ,σr,
,
+α–β–μ–
s
j=
λjkj;σ, . . . ,σr,
and
D=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–γ+α+α–
s
j=
λjkj;σ, . . . ,σr,
–μ–γ +α+β–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–β–
s
j=
λjkj;σ, . . . ,σr,
.
Proof For convenience, let the left-hand side of (.) be denoted by I. Applying the Marichev-Saigo-Maeda generalized fractional operators (.) to the LHS of (.) and mak-ing use of (.) and (.), we obtain
I=
Iα+,α,β,β,γ
tμ–(b–at)–γ
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)
×ctλ(b–at)δ
k
· · ·cstλs(b–at)δs
ks×
(πi)r
L · · ·
Lr
φ(ξ, . . . ,ξr)
× r
i= θi(ξi)
zitσi(b–at)νi
ξi
dξ· · ·dξr
(x). (.)
By changing the order of summation and integration, we have
I=
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×
(πi)r
L · · ·
Lr
φ(ξ, . . . ,ξr) r
i=
θi(ξi)zi ξi
I+α,α,β,β,γtμ+
s
j=λjkj+ri=σiξi–
×(b–at)–η–
s
j=λjkj–ri=σiξi (x)dξ
· · ·dξr
. (.)
By using the binomial expansion for (b–at)–η–
s j=λjkj–
r
i=σiξi and applying the
Mellin-Barnes counter integral, we get
I=
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckssb –sj=λjkj
×
(πi)r
L · · ·
Lr
φ(ξ, . . . ,ξr)× r
i=
θi(ξi)zib– r
i=σiξi ξi
×
Lr+
(η+sj=λjkj+
r
i=σiξi+ξr+) (η+sj=λjkj+ri=σiξi)(ξr++ ) ×I+α,α,β,β,γtμ+
s j=λjkj+
r
i=σiξi+ξr+– (x)dξ
· · ·dξr
Applying (.) to (.) and re-interpreting the Mellin-Barnes counter integral in terms of the multivariableH-function ofr+ variables, we obtain the right-hand side of (.) after a few simplifications.
Assertion (.) of Theorem can be proved in a similar manner by using (.) of
Lemma . So, we omit the details of the proof of Theorem .
Here, we derive certain, presumably, new formulas involving Marichev-Saigo-Maeda type fractional integral operators. By settingn=p=q= in (.) and (.), respectively, we obtain two fractional integral formulas involving product of ther,H-functions stated in Corollaries and below.
Corollary Letα,α,β,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . .r};j∈ {, . . . ,s})
and (γ) > .Then the following relation holds true:
Iα+,α,β,β,γ
tμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
× r
i=
Hmi,ni
pi,qi
zitσi(b–at)–vi
(cij,Cji),pi
(dij,Dij),qi
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s j=δjkj
x
s j=λjkj
H,:mi,ni;,
,:pi,qi;,
zix
σi
bvi –a
bx
A: (c(r)j ,γ (r) j ),pi; – B: (dj(r),δ(r)j ),qi; (, )
, (.)
where A and B are given in Theoremand Hm,n
p,q (·)is the familiar Fox H-function.
Corollary Letα,α,β,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . . ,r};j∈ {, . . . ,s})
and (γ) > .Then the following result holds true:
Iα–,α,β,β,γ
tμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
× r
i=
Hmi,ni
pi,qi
zitσi(b–at)–vi
(ci j,Cji),pi
(di j,Dij),qi
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s
j=δjkjxsj=λjkjH,:mi,ni;,
,:pi,qi;,
zix
σi
bvi
–a bx
C: (c(r)j ,γ (r) j ),pi; – D: (dj(r),δj(r)),qi; (, )
, (.)
where
C=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–
s
j=
λjkj;σ, . . . ,σr,
–μ–γ +α+α+β–
s
j=
λjkj;σ, . . . ,σr,
,
–μ+α–β–
s
j=
λjkj;σ, . . . ,σr,
and
D=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–γ +α+α–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–γ+α+β–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–β–
s
j=
λjkj;σ, . . . ,σr,
,
where Hm,n
p,q (·)is the familiar Fox H-function.
Interestingly, in view of relation (.), we arrive at the following corollaries concerning Saigo fractional integral operator. It is similar to the results due to Agarwal [], p. and p., Eqs. () and ().
Corollary Letα,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . . ,r};j∈ {, . . . ,s}),and
(α) > .Let the following conditions be satisfied,too: (i) |argzi|<π
i,i> (i= , . . . ,r),where
i= – p
j=n+ αj(i)–
q
j= βj(i)+
ni
j= γj(i)–
pi
j=ni+
γj(i)
+
mi
j= δj(i)–
qi
j=mi+
δj(i)> i∈ {, . . . ,r}.
(ii)
(μ) +
r
i= σi min
≤j≤mi
(d(i) j ) δj(i)
>max, (β–γ),
(η) +
r
i=
vi min ≤j≤mi
(d(i) j ) δj(i)
>max, (β–γ) and a
bx
< .
Then the following result holds:
Iα+,β,γtμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×Hztσ(b–at)–v, . . . ,zrtσr(b–at)–vr (x)
=b–ηxμ–β–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
×b–
s j=δjkjx
s
j=λjkjH,n+:{mr,nr};,
p+,q+:{pr,qr};,
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
zx
σ bv
.. . zrx
σr
bvr
–abx
(aj;α()j , . . . ,α (r)
j ),p,A:{(c(r)j ,γ (r) j ),pr}; –
(bj;βj(), . . . ,β (r)
j ),q,B:{(dj(r),δ (r)
j ),qr}; (, )
⎤ ⎥ ⎥ ⎥ ⎥
⎦, (.)
where
A=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–
s
j=
λjkj;σ, . . . ,σr,
,
–μ–γ +β–
s
j=
λjkj;σ, . . . ,σr,
and
B=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ+β–
s
j=
λjkj;σ, . . . ,σr,
–μ–γ –α–
s
j=
λjkj;σ, . . . ,σr,
,
and the conditions of existence of the above corollary follow easily from Theorem.
Corollary Letα,β,γ,μ,η,δj,vi,zi,a,b,cj∈C,λj,σi> (i∈ {, . . . ,r};j∈ {, . . . ,s}),and
(α) > .Let the following conditions be satisfied,too: (i) |argzi|<πi,i> (i= , . . . ,r),where
i= – p
j=n+ αj(i)–
q
j= βj(i)+
ni
j= γj(i)–
pi
j=ni+
γj(i)+
mi
j= δj(i)–
qi
j=mi+
δj(i)> .
(ii)
(μ) –
r
i= σi min
≤j≤mi
(d(i) j ) δ(i)j
< +max (β), (γ) (i= , . . . ,r),
(η) –
r
i=
vi min ≤j≤mi
(d(i) j ) δj(i)
< +max (β), (γ) (i= , . . . ,r).
Then the following result holds:
Iα–,β,γtμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×Hztσ(b–at)–v, . . . ,zrtσr(b–at)–vr (x)
=b–ηxμ–β–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
×b–
s j=δjkjx
s
j=λjkjH,n+:{mr,nr};,
p+,q+:{pr,qr};,
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
zx
σ bv
.. . zrx
σr
bvr
–abx
(aj;αj(), . . . ,α (r)
j ),p,C:{(c(r)j ,γ (r) j ),pr}; –
(bj;βj(), . . . ,β (r)
j ),q,D:{(d(r)j ,δ (r)
j ),qr}; (, )
⎤ ⎥ ⎥ ⎥ ⎥
⎦, (.)
where
C=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–μ–
s
j=
λjkj;σ, . . . ,σr,
,
+α+β–μ–γ –
s
j=
λjkj;σ, . . . ,σr,
and
D=
–η–
s
j=
δjkj;v, . . . ,vr,
,
–γ –μ–
s
j=
λjkj;σ, . . . ,σr,
+β–μ–
s
j=
λjkj;σ, . . . ,σr,
,
and the conditions of existence of the above corollary follow easily from Theorem.
Remark By puttingβ= –αandβ= in Corollaries and , respectively, we can ob-tain very interesting results involving the Riemann-Liouville and Erdélyi-Kober fractional integral operators.
3 Special cases and concluding remarks
Here we consider another variation of the results derived in the preceding sections. The multivariableH-function occurring in these results can be suitably specialized to yield a wide variety of special functions (or product of such functions) of one or more variables (see [], pp.-, -, - and []). Again, by suitably specializing the coefficient of the first class of multivariable polynomials, it can be reduced to other multivariable hy-pergeometric polynomials and classical orthogonal polynomials of one or more variables. (i) If we reduce the multivariableH-functions into the product of two FoxH-functions in Theorem and then reduce oneH-function to the exponential function by takingσ= ,
v→, we obtain the following result after a little simplification:
Iα+,α,β,β,γ
tμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×e–ztHp,qm,n
ztσ(b–at)–v
(cj,Cj),p
(dj,Dj),q
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
×b–
s j=δjkjx
s
j=λjkjH,:,;m,n;,
,:,;p,q;,
⎡ ⎢ ⎣
zx
zx
σ
bv
–abx
E: (cj,γj),p; –
F: (, );{(dj,δj),q}; (, )
⎤ ⎥
⎦, (.)
where
E=
–η–
s
j=
δjkj; ,v,
,
–μ–
s
j=
λjkj; ,σ,
,
–μ–γ +α+α+β–
s
j=
λjkj; ,σ,
,
–μ+α–β–
s
j=
λjkj; ,σ,
,
F=
–η–
s
j=
δjkj; ,v,
,
–μ–γ +α+α–
s
j=
λjkj; ,σ,
,
–μ–γ +α+β–
s
j=
λjkj; ,σ,
,
–μ–β–
s
j=
λjkj; ,σ,
.
The conditions of validity of the above result easily follow from Theorem .
Remark If we reduce the Marichev-Saigo-Maeda type fractional integral operator into
the Riemann-Liouville fractional integral operator and putSm,...,ms
n = ,η,v= , and make
a suitable adjustment in the parameters in the above equation, then we obtain the known result by Kilbas and Saigo [], p., Eq. (..); in addition, if we putSmnjj = ,η,v= , z= /,σ= and reduce the FoxH-function to the Bessel function of the first kind,
then we obtain the known result given by Kilbas and Sebastian [], p., Eqs. ()-().
(ii) Next if we takez,σ= andv= in the result (.) and reduce theH-function
of one variable to a generalized Mittag-Leffler function [] (see also []), we can easily obtain the following result after a few simplifications:
Iα+,α,β,β,γtμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×e–ztEvω,ξ[t] (x)
=b
–ηxμ–α–α+γ–
(v)
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s
j=δjkj(–x)sj=λjkjH,:,;,;, ,:,;,;,
⎡ ⎢ ⎣
zx
x –a bx
E: –; ( –v, ); –
F: (, ); (, ), ( –η; ); ( –ξ,ω); (, )
⎤ ⎥
where
E=
–η–
s
j=
δjkj; , ,
,
–μ–
s
j=
λjkj; , ,
,
–μ–γ +α+α+β–
s
j=
λjkj; , ,
,
–μ+α–β–
s
j=
λjkj; , ,
,
F=
–η–
s
j=
δjkj; , ,
,
–μ–γ+α+α–
s
j=
λjkj; , ,
,
–μ–γ +α+β–
s
j=
λjkj; , ,
,
–μ–β–
s
j=
λjkj; , ,
.
The conditions of validity of (.) can be easily followed directly by those given with (.).
Remark If we reduce the Marichev-Saigo-Maeda type fractional integral operator to
the Riemann-Liouville fractional integral operator and setSm,...,ms
n = ,η,v= , and make
a suitable adjustment in the parameters in equation (.), then we arrive at the known result given by Saxenaet al.[], p., Eq. (.).
(iii) Again, if we reduce theH-function of one variable to the generalized Wright hyper-geometric function [], p., Eq. (..), in the result (.), we arrive at the following new and interesting result:
Iα+,α,β,β,γ
tμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
×e–ztpψq
–ztσ(b–at)–v
( –cj,Cj),p
(, ), ( –dj,Dj),q
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s
j=δjkjxjs=λjkjH,:,;,p;,
,:,;p,q;,
⎡ ⎢ ⎣
zx
–zx
σ bv
–abx
E: (cj,Cj),p; –
F: (, );{(dj,Dj),q}; (, )
⎤ ⎥
⎦, (.)
where
E=
–η–
s
j=
δjkj; ,v,
,
–μ–
s
j=
λjkj; ,σ,
,
–μ–γ +α+α+β–
s
j=
λjkj; ,σ,
,
–μ+α–β–
s
j=
λjkj; ,σ,
F=
–η–
s
j=
δjkj; ,v,
,
–μ–γ +α+α–
s
j=
λjkj; ,σ,
,
–μ–γ +α+β–
s
j=
λjkj; ,σ,
,
–μ–β–
s
j=
λjkj; ,σ,
.
(iv) Further Theorem , if we reduce the multivariableH-function to the product of r-different Whittaker functions [], p., Eq. (..), and takevi→,σi= , then we
arrive at the following result:
Iα+,α,β,β,γ
tμ–(b–at)–ηSm,...,ms
n
ctλ(b–at)–δ, . . . ,cstλs(b–at)–δs
× r
i=
e–zit Wλ
i,μi(zit)
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s j=δjkjx
s
j=λjkjH,:,;...;,;,
,:,;...;,;,
⎡ ⎢ ⎢ ⎢ ⎢ ⎣
zx
.. . zrx
–abx
G: –; ( –λ, ); . . . ; ( –λr, ); –
H: (/±μ, ); . . . ; (/±μr, )(, )
⎤ ⎥ ⎥ ⎥ ⎥
⎦, (.)
where
G=
–η–
s
j=
δjkj; , . . . , ! rtimes
,
,
–μ–
s
j=
λjkj; , . . . , ! rtimes
,
,
–μ–γ +α+α+β–
s
j=
λjkj; , . . . , ! rtimes
,
,
–μ+α–β–
s
j=
λjkj; , . . . , ! rtimes
,
,
H=
–η–
s
j=
δjkj; , . . . , ! rtimes
,
,
–μ–γ +α+α–
s
j=
λjkj; , . . . , ! rtimes
,
,
–μ–γ +α+β–
s
j=
λjkj; , . . . , ! rtimes
,
,
–μ–β–
s
j=
λjkj; , . . . , ! rtimes
,
.
The conditions of validity of the above result easily follow from Theorem .
Remark If we reduce the Marichev-Saigo-Maeda type fractional integral operator to the Riemann-Liouville fractional integral operator and setSm,...,ms
n = ,η,v= , and make
(v) Finally, if we reduce the multivariable H-function into the product of two Fox
H-functions in Theorem and then reduce oneH-function to the exponential function
by settingσ,λj,v,δj= ,cj,s= andSmn to the Hermite polynomial [, ], and we set
S
n[x] =xn/Hn[√x], in this casem= ,An,k= (–)k, we obtain the following interesting
result:
Iα+,α,β,β,γ
tμ–(b–at)–ηtn/Hn
√t
×e–ztHp,qm,n
ztσ(b–at)–v
(cj,Cj),p
(dj,Dj),q
(x)
=b–ηxμ–α–α+γ–
mk+···+msks≤n
k,...,ks=
(–n)mk+···+msks k!· · ·ks!
(n;k, . . . ,ks)ck · · ·ckss
×b–
s j=δjkjx
s
j=λjkjH,:,;{m,n};,
,:,;{p,q};,
⎡ ⎢ ⎣
zx
zx
σ bv
–abx
G: (c(r)j ,γj(r)),pr; – H: (, );{(dj,δj),q}; (, )
⎤ ⎥
⎦, (.)
where
G=
–η–
s
j=
δjkj; ,v,
,
–μ–
s
j=
λjkj; ,σ,
,
–μ–γ +α+α+β–
s
j=
λjkj; ,σ,
,
–μ+α–β–
s
j=
λjkj; ,σ,
,
H=
–η–
s
j=
δjkj; ,v,
,
–μ–γ +α+α–
s
j=
λjkj; ,σ,
,
–μ–γ +α+β–
s
j=
λjkj; ,σ,
,
–μ–β–
s
j=
λjkj; ,σ,
.
The conditions of validity of the above result can be easily derived from Theorem . A large number of other special cases of our main result can also be obtained, but we do not mention them here on account of the lack of space.
Now, we conclude our present investigation by remarking that the fractional integration (of Marichev-Saigo-Maeda type) of the products of multivariableH-functions and the first class of multivariable polynomials established in this paper will be useful for investigators in various disciplines of applied sciences and engineering physics. We are also trying to find certain possible applications of those results presented here to some other research areas, for example, obtaining a closed form solution of a fractional generalization of a free electron equation.
Competing interests
Authors’ contributions
All authors have participated in the obtained results. The collaboration of each one cannot be separated in different parts of the paper. All of them have made substantial contributions to the theoretical results. All authors have been involved in drafting the manuscript and revising it critically for important intellectual content. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Anand International College of Engineering, Jaipur, 303012, India.2Department of
Economics, Belarusian State University, Minsk, 220030, Belarus.3Institute of Mathematics, National University of
Uzbekistan, Tashkent, 100125, Uzbekistan.4Department of Mathematics, Fateh College for Women, Bathinda, 151103,
India.
Acknowledgements
We are grateful to Dr. S Jain for her useful suggestions during discussions and to anonymous reviewers for their valuable remarks.
Endnote
a Here and in what follows, we denote byωan imaginary unit.
Received: 22 July 2015 Accepted: 30 October 2015
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