Fractional calculus of a unified Mittag-Leffler function
The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions L(a, b). Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function E α,β,λ,μ,ρ,p γ,δ (cz; s, r) are studied and the integral...
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irk-123456789-1660962020-02-19T01:25:04Z Fractional calculus of a unified Mittag-Leffler function Prajapati, J.C. Nathwani, B.V. Статті The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions L(a, b). Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function E α,β,λ,μ,ρ,p γ,δ (cz; s, r) are studied and the integral representations are obtained. Some properties of a special case of this function are also studied by the means of fractional calculus. Головною метою роботи є введення оператора у просторі L(a,b) дійсних або комплексних Функцій, вимірних відносно міри Лебега. Вивчено деякі властивості дробових інтегралів Рімана-Ліувілля та диференціальних операторів, що відповідають функції Eγ,δα,β,λ,μ,ρ,p(cz;s,r). Отримано відповідні інтегральні зображення. Деякі властивості частинного випадку цієї функції також вивчено за допомогою дробового числення. 2014 Article Fractional calculus of a unified Mittag-Leffler function / J.C. Prajapati, B.V. Nathwani // Український математичний журнал. — 2014. — Т. 66, № 8. — С. 1133–1145. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166096 517.9 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Prajapati, J.C. Nathwani, B.V. Fractional calculus of a unified Mittag-Leffler function Український математичний журнал |
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The main aim of the paper is to introduce an operator in the space of Lebesgue measurable real or complex functions L(a, b). Some properties of the Riemann–Liouville fractional integrals and differential operators associated with the function E α,β,λ,μ,ρ,p γ,δ (cz; s, r) are studied and the integral representations are obtained. Some properties of a special case of this function are also studied by the means of fractional calculus. |
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Article |
| author |
Prajapati, J.C. Nathwani, B.V. |
| author_facet |
Prajapati, J.C. Nathwani, B.V. |
| author_sort |
Prajapati, J.C. |
| title |
Fractional calculus of a unified Mittag-Leffler function |
| title_short |
Fractional calculus of a unified Mittag-Leffler function |
| title_full |
Fractional calculus of a unified Mittag-Leffler function |
| title_fullStr |
Fractional calculus of a unified Mittag-Leffler function |
| title_full_unstemmed |
Fractional calculus of a unified Mittag-Leffler function |
| title_sort |
fractional calculus of a unified mittag-leffler function |
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Інститут математики НАН України |
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2014 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/166096 |
| citation_txt |
Fractional calculus of a unified Mittag-Leffler function / J.C. Prajapati, B.V. Nathwani // Український математичний журнал. — 2014. — Т. 66, № 8. — С. 1133–1145. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT prajapatijc fractionalcalculusofaunifiedmittaglefflerfunction AT nathwanibv fractionalcalculusofaunifiedmittaglefflerfunction |
| first_indexed |
2025-07-14T20:45:29Z |
| last_indexed |
2025-07-14T20:45:29Z |
| _version_ |
1837656629589311488 |
| fulltext |
UDC 517.9
J. C. Prajapati (Charotar Univ. Sci. and Technology, Changa, India),
B. V. Nathwani (Sardar Vallabhbhai Patel Inst. Technology, Vasad, India)
FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION
ДРОБОВЕ ЧИСЛЕННЯ УНIФIКОВАНОЇ ФУНКЦIЇ МIТТАГ-ЛЕФФЛЕРА
The main aim of thе paper is to introduce an operator in the space of Lebesgue measurable real or complex functions
L(a, b). Certain properties of the Riemann – Liouville fractional integrals and differential operators associated with the
function Eγ,δ
α,β,λ,µ,ρ,p(cz; s, r) are studied and the integral representations are obtained. Some properties of a special case
of this function are also studied by means of fractional calculus.
Головною метою роботи є введення оператора у просторi L(a, b) дiйсних або комплексних функцiй, вимiрних вiд-
носно мiри Лебега. Вивчено деякi властивостi дробових iнтегралiв Рiмана – Лiувiлля та диференцiальних операторiв,
що вiдповiдають функцiї Eγ,δ
α,β,λ,µ,ρ,p(cz; s, r). Отримано вiдповiднi iнтегральнi зображення. Деякi властивостi
частинного випадку цiєї функцiї також вивчено за допомогою дробового числення.
1. Introduction, definitions and preliminaries. The Mittag-Leffler function has been studied by
many researchers either in context with obtaining new properties or by introducing a new generaliza-
tion and then deriving its properties [9, 11, 13]. Recently, we [7] have also studied various properties
of our newly introduced generalization of Mittag-Leffler function in the form
Eγ,δα,β,λ,µ,ρ,p(cz; s, r) =
∞∑
n=0
[(γ)δn]s (cz)(pn+ρ−1)
Γ(α(pn+ ρ− 1) + β) [(λ)µn]r (ρ)pn
, (1.1)
where α, β, γ, λ, ρ ∈ C, Re(α, β, γ, λ, ρ) > 0; δ, µ, p, c > 0 and (γ)q n =
Γ(γ + q n)
Γ(γ)
is the
generalized Pochhammer symbol [8]. In particular, if q ∈ N. it takes the form
(γ)q n = qqn
q∏
r=1
(
γ + r − 1
q
)
n
.
If p = 1, ρ = 1, r = 0, s = 1, δ = q, s = 1, c = 1, then (1.1) yields the generalization due to Shukla
and Prajapati [11]. Here, we also introduce an operator denoted and defined by
(
Eγ,δα,β,λ,µ,ρ,p,ω;a+f
)
(x) =
x∫
a
(x− t)β−1 Eγ,δα,β,λ,µ,ρ,p
(
ω(x− t)α; s, r
)
f(t) dt, (1.2)
where α, β, γ, λ, ρ, ω ∈ C; Re(α, β, γ, λ, ρ) > 0; δ, µ, p > 0, and x > a.
We enlist the following definitions and well-known formulas for studying the properties of
the Riemann – Liouville (R–L) fractional integrals and differential operators associated with our
generalization (1.1) as well as as the operator (1.2).
The space L(a, b) of (real or complex valued) Lebesgue measurable functions [4, 10] is given by
L(a, b) =
f : ‖f‖1 =
b∫
a
|f(t)|dt <∞
. (1.3)
c© J. C. PRAJAPATI, B. V. NATHWANI, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8 1133
1134 J. C. PRAJAPATI, B. V. NATHWANI
For f(x) ∈ L(a, b), µ ∈ C, and Re(µ) > 0, the R–L fractional integrals of order µ [10] are defined
as follows.
The left-sided R–L fractional integral operator of order µ is defined as
aI
µ
x f(x) = Iµa+f(x) =
1
Γ(µ)
x∫
a
f(t)
(x− t)1−µ
dt, x > a, (1.4)
where the right-sided R–L fractional integral operator of order µ is defined as
xI
µ
b f(x) := Iµb−f(x) =
1
Γ(µ)
b∫
x
f(t)
(x− t)1−µ
dt, x < b. (1.5)
Further, if µ, β ∈ C, Re(µ, β) > 0, then [6, 10]
Iµa+[(t− a)β−1](x) =
Γ(β)
Γ(µ+ β)
(x− a)µ+β−1. (1.6)
For µ ∈ C, Re(µ) > 0; n = [Re(µ)] + 1, the R–L fractional derivative is
(Dα
a+ f)(x) =
(
d
dx
)n
(In−αa+ f)(x). (1.7)
Then for α, β, γ, λ, ρ,∈ C, Re(α, β, γ, λ, ρ, (β − m)) > 0, and δ, µ, p,m ∈ N, we have shown
that [7] (
d
dz
)m [
zβ−1 Eγ,δα,β,λ,µ,ρ,p
(
ω (cz)α; s, r
)]
=
= zβ−m−1 Eγ,δα,β−m,λ,µ,ρ,p
(
ω (cz)α; s, r
)
. (1.8)
The fractional integral operator investigated by Erdélyi – Kober is defined and represented as
Iη,νx {f(x)} =
x−η−ν+1
Γ(ν)
x∫
0
(x− t)ν−1f(t) dt, Re(ν) > 0, η > 0, (1.9)
which is a generalization of the R–L fractional integral operator (1.5).
Hilfer [2, 3] generalized the R–L fractional derivative operator Dµ
a+ in (1.6) by introducing a
right-sided fractional derivative operator Dµ,ν
a+ of order 0 < µ < 1 and type 0 ≤ ν ≤ 1 with respect
to x as follows:
(Dµ,ν
a+ f)(x) =
(
I
ν(1−µ)
a+
d
dx
(I
(1−ν)(1−µ)
a+ f)
)
(x). (1.10)
The difference between the fractional derivatives of various types becomes apparent from the follow-
ing formula involving the Laplace transformation [2, 3]:
L[Dµ,ν
0 f(x)](s) = sµ L[f(x)](s)− sν(1−mu)
(
I
(1−ν)(1−µ)
0 f
)
(0+), (1.11)
where 0 < µ < 1, and the initial-value term:
(
I
(1−ν)(1−µ)
0 f
)
(0+) involves the R–L fractional integral
operator of order (1− ν)(1− µ) evaluated in the limit as t→ 0+. Here, as usual
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1135
L[f(x)](s) =
∞∫
0
e−sx f(x) dx, (1.12)
provided that the defining integral exists.
Prajapati, Dave and Nathwani [7] has shown that the Mellin – Barnes integral for the function
defined by (1.1) is given by
Eγ,δα,β,λ,µ,ρ,p (z; s, r) =
[Γ(λ)]r Γ(ρ) p zρ−1
2πi [Γ(γ)]s
×
×
∫
L
Γ(−pξ) Γ(1 + pξ) [Γ(γ + δξ)]s (−z)pξ
Γ(β + αρ− α+ αpξ) [Γ(λ+ µξ)]r Γ(ρ+ pξ)
dξ. (1.13)
Wright generalized hypergeometric function [1] is defined as
pψq
[
(a1, A1), . . . , (ap, Ap); z
(b1, B1), . . . , (bq, Bq);
]
=
∞∑
r=0
∏p
j=1
Γ(aj + rAj)∏q
j=1
Γ(bj + rBj)
zr
r!
(1.14)
= H1,p
p,q+1
−z (1− a1, A1), . . . , (1− ap, Ap)
(0, 1), (1− b1, B1), . . . , (1− bq, Bq)
, (1.15)
where Hm,n
p,q
[
−z
(a1, A1), . . . , (ap, Ap)
(b1, B1), . . . , (bq, Bq)
]
denotes the Fox H-function and ai, bj ∈ C, Ai, Bj ∈ R,
i = 1, 2, . . . , p; j = 1, 2, . . . , q, 1 +
∑q
j=1
Bj −
∑p
i=1
Ai > 0.
2. Main results. We prove in this section the following results.
Theorem 2.1. Let a ∈ R+ = [0,∞), α, β, γ, λ, ρ, η ∈ C, Re(α, β, γ, λ, ρ, η) > 0; δ, µ, p > 0
for x > a, then (
Iηa+ (t− a)β−1 Eγ,δα,β,λ,µ,ρ,p(ω(c(t− a))α; s, r
)
(x) =
= (x− a)(η+β−1) Eγ,δα,β+η,λ,µ,ρ,p(ω(c(x− a))α; s, r) (2.1)
and (
Dη
a+ (t− a)β−1 Eγ,δα,β,λ,µ,ρ,p(ω(c(t− a))α; s, r)
)
(x) =
= (x− a)(β−η−1) Eγ,δα,β−η,λ,µ,ρ,p(ω(c(x− a))α; s, r). (2.2)
Proof. Applying (1.1) to the left-hand side of (1.13) and then using (1.6), it yields(
Iηa+ (t− a)β−1 Eγ,δα,β,λ,µ,ρ,p(ω(c(t− a))α; s, r
)
(x) =
= (x− a)β+η−1
∞∑
n=0
[(γ)δn]s(ω (c (x− a))α)(pn+ρ−1)
Γ(α(pn+ ρ− 1) + β + η) [(λ)µn]r (ρ)pn
.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1136 J. C. PRAJAPATI, B. V. NATHWANI
Here using (1.1) once again leads us to (2.1).
Now, using (1.7) to the the left-hand side of (2.2) and then applying (2.1), we get(
Dη
a+ (t− a)β−1 Eγ,δα,β,λ,µ,ρ,p(ω(c(t− a))α; s, r)
)
(x) =
=
(
d
dx
)n[
(x− a)β−η−1 Eγ,δα,β−η,λ, µ,ρ,p(ω(c(x− a))α; s, r)
]
.
Further use of (1.7), gives the proof of (2.2).
Theorem 2.2. Let α, γ, λ, ρ, η ∈ C; Re(α),Re(γ),Re(λ),Re(ρ),Re(η) > 0; δ, µ, p > 0, then
0I
η
x
[
ν E1,δ
α,1,λ,µ,ρ,p(ν(cx)α; s, r)
]
= ν xη E1,δ
α,η+1, λ,µ,ρ,p(ν(cx)α; s, r). (2.3)
Proof. Applying (1.4) to the left-hand side of (2.3) and then using (1.1), we get
0I
η
x
[
ν E1,δ
α,1,λ,µ,ρ,p(ν(cx)α; s, r)
]
=
=
1
Γ(η)
∞∑
n=0
[(1)δn]s cα(pn+ρ−1)ν(pn+ρ)
Γ(α(pn+ ρ− 1) + 1) [(λ)µn]r (ρ)pn
x∫
0
tα(pn+ρ−1)(x− t)(η−1) dt.
After some simplification and further use of (1.1), gives the proof of (2.3).
Theorem 2.3. Let α, β, γ, λ, ρ, ν, ω ∈ C; Re(α, β, γ, λ, ρ, ν) > 0; δ, µ, p > 0, then(
Eγ,δα,β,λ,µ,ρ,p,ω;a+ (t− a)ν−1
)
(x) = (x− a)β+ν−1 Γ(ν)Eγ,δα,β+ν, λ,µ,ρ,p
(
ω(x− a)α
)
. (2.4)
Proof. Putting f(t) = (t− a)ν−1 in (1.2), we get
(
Eγ,δα,β,λ,µ,ρ,p,ω;a+ (t− a)ν−1
)
(x) =
x∫
a
(x− t)β−1 Eγ,δα,β,λ,µ,ρ,p,
(
ω(x− t)α
)
(t− a)ν−1 dt,
and using (1.1), this reduced to
=
∞∑
n=0
[(γ)δ n]s ω(p n+ρ−1)
Γ(α(p n+ ρ− 1) + β) [(λ)µ n]r (ρ)pn
×
×
x∫
a
(x− t)α(p n+ρ−1)+β−1 (t− a)ν−1 dt
and simplifying the above equation, it becomes
=
∞∑
n=0
[(γ)δ n]s ((ρ)pn)−1 ω(p n+ρ−1)
Γ(α(p n+ ρ− 1) + β)[(λ)µ n]r
B(α(p n+ ρ) + β − 1, ν)
and further simplification of the above equation gives the proof of (2.4).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1137
We show that the operator defined (1.2) is in fact bounded; whose proof is given below in the
form of the following theorem.
Theorem 2.4. Let the function φ be in the space L(a, b) of Lebesgue measurable functions on a
finite interval [a, b] of the real line R given by
L(a, b) =
f : ‖f‖1 =
b∫
a
|f(t)|dt <∞
.
Then the integral operator Eω;γ,δa+; α,β,λ,µ,ρ,p is bounded on L(a, b) and∥∥∥Eγ,δα,β,λ,µ,ρ,p,ω;a+φ∥∥∥
1
≤M‖φ‖1, (2.5)
where the constant M, 0 <M <∞, given by
M = (b− a)Re(β)
∞∑
k=0
|(γ)δ k|s
|Γ(α (pk + ρ− 1) + β)| (Re (α(p k + ρ− 1) + β))
×
×|ω ((b− a)c)α|Re(pk+ρ−1)
|(λ)µ k|r |(ρ)pk|
. (2.6)
Proof. Using (1.2) and (1.3) and interchanging the order of integration by applying the Dirichlet
formula [9], we have ∥∥∥Eγ,δα,β,λ,µ,ρ,p,ω;a+φ∥∥∥
1
=
=
b∫
a
∣∣∣∣∣∣
x∫
a
(x− t)β−1 Eγ,δα,β,λ,µ,ρ,p( ω(c(x− t))α; s, r)) φ(t) dt
∣∣∣∣∣∣ dx ≤
≤
b∫
a
b∫
t
(x− t)Re(β)−1
∣∣∣Eγ,δα,β,λ,µ,ρ,p(ω(c(x− t))α; s, r)
∣∣∣ dx
|φ(t)| dt.
On substituting x− t = u, using (1.1) and simplification of the above equation yields
=
b∫
a
b−t∫
0
uRe(β)−1
∣∣∣Eγ,δα,β,λ,µ,ρ,p(ω(cu)α; s, r)
∣∣∣ du
|φ(t)| dt ≤
≤
b∫
a
∞∑
k=0
|(γ)δ k|s (ω cα)pk+ρ−1∣∣Γ(α (pk + ρ− 1) + β)
∣∣ |(λ)µk|r |(ρ)pk|
×
×
b−a∫
0
uRe(α(pk+ρ−1)+β−1)
|φ(t)| dt =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1138 J. C. PRAJAPATI, B. V. NATHWANI
=
b∫
a
∞∑
k=0
|(γ)δ k|s (ω cα)pk+ρ−1|b− a|Re(α(pk+ρ−1)+β) |φ(t)|∣∣Γ(α (pk + ρ− 1) + β)
∣∣ |(λ)µk|r |(ρ)pk|Re (α(pk + ρ− 1) + β)
dt =
= (b− a)Re(β)
∞∑
k=0
|(γ)δ k|s
|Γ(α (pk + ρ− 1) + β)| |(λ)µ k|r |(ρ)pk|
×
×|ω (c(b− a))Re (α)|pk+ρ−1
Re (α(pk + ρ− 1) + β)
b∫
a
|φ(t)| dt =
= (b− a)Re(β)
∞∑
k=0
|(γ)δ k|s |ω ((b− a)c)α|Re(pk+ρ−1)
|Γ(α (pk + ρ− 1) + β)| Re (α(pk + ρ− 1) + β)
‖φ‖1
|(λ)µ k|r |(ρ)pk|
=
= M ‖φ‖1,
where M is finite and given by (2.6). This completes proof of the boundedness property of the
integral operator Eγ,δα,β,λ,µ,ρ,p,ω;a+ as asserted by Theorem 2.4.
The following theorem incorporates the fractional differential equation for (1.2).
Theorem 2.5. If 0 < η < 1, 0 ≤ ν ≤ 1, ω, ξ ∈ C, R(α) = R(δ)− 1 > 0 and min{Re(β, γ, λ,
µ, ρ)} > 0, then (
Dη,ν
0+ y
)
(x) = ξ
(
Eγ,δα,β,λ,µ,ρ,p,ω;0+
)
(x) + f(x) (2.7)
with the initial condition (
I0 +(1−ν)(1−η) y
)
(0+) = C,
has solution in the space L(0,∞) given by
y(x) = C
xη−ν(1−η)−1
Γ(η − ν + ην)
+ ξ xη+β Eα,β+η+1,λ,µ,ρ,p
(
ω(a xα)
)
+
+
1
Γ(η)
x∫
0
(x− t)η−1 f(t) dt, (2.8)
where C is arbitrary constant.
Proof. Applying the Laplace transform of each side of (2.7), and using the formulas (1.2) and
(1.11), we find by means of the Laplace convolution theorem that
sη Y (s)− C sν (1−η) = ξ L
[
xβ−1 Eγ,δα,β,λ,µ,ρ,p(ω x
α)
]
(s)L(1)(s) + F (s) =
= ξ s−β−1
∞∑
n=0
[(γ)δ n]s (ω (as)α)pn+ρ−1
[(λ)µ n]r(ρ)pn
+ F (s)
which readily yields
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FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1139
Y (s) = C sν (1−η)−η + ξ s−β−η−1
∞∑
n=0
[(γ)δ n]s (ω (as)α)pn+ρ−1
[(λ)µ n]r (ρ)pn
+ F (s) s−η. (2.9)
Now, by taking the inverse Laplace transform of each side of equation (2.9), we get
y(x) = C L−1
(
sν(1−η)−η
)
(x)+
+ξ
∞∑
n=0
[(γ)δn]s (ω (a)α)pn+ρ−1
[(λ)µn]r (ρ)pn
L−1(s−α(pn+ρ−1)−β−η−1)(x) + L−1(s−η F (s)) =
= C
xη−ν(1−η)−1
Γ(η − ν + ην)
+ ξ xη+β
∞∑
n=0
[(γ)δn]s (ω aα)pn+ρ−1 [(ρ)pn]−1 xα(pn+ρ−1)
Γ(α(pn+ ρ− 1) + β + η + 1)[(λ)µn]r
+
+
1
Γ(η)
x∫
0
(x− t)η−1 f(t) dt =
= C
xη−ν(1−η)−1
Γ(η − ν + η ν)
+ ξ xη+β Eγ,δα,β+η+1,λ,µ,ρ,p
(
ω(a xα)
)
+
+
1
Γ(η)
x∫
0
(x− t)η−1 f(t) dt
which completes the proof of Theorem 2.5 under the various already stated parametric constraints.
Theorem 2.6. If 0 < η < 1, 0 ≤ ν ≤ 1, ω, ξ ∈ C, R(α) = R(δ) − 1 > 0 and min{R(β, γ,
λ, µ, ρ)} > 0, then(
Dη,ν
0+ y
)
(x) = ξ
(
Eγ,δα,β,λ,µ,ρ,p,ω;0+
)
(x) + xβ Eγ,δα,β+1,λ,µ,ρ,p
(
(ω(ax)α); s, r
)
(2.10)
with the initial condition (
I0 +(1−ν)(1−η) y
)
(0+) = C
has solution in the space L(0,∞) given by
y(x) = C
xη−ν(1−η)−1
Γ(η − ν + ην)
+ (ξ + 1) xη+β Eγ,δα,βη+1,λ,µ,ρ,p
(
(ω(ax)α); s, r
)
, (2.11)
where C is arbitrary constant.
Proof. Now, substituting
f(t) = tβ Eγ,δα,β+1,λ,µ,ρ,p
(
(ω(at)α); s, r
)
in above Theorem 2.5, we get
y(x) = C
xη−ν(1−η)−1
Γ(η − ν + η ν)
+ ξ xη+β Eγ,δα,β+η+1,λ,µ,ρ,p
(
(ω(ax)α); s, r
)
+
+
1
Γ(η)
x∫
0
(x− t)η−1 tβ Eγ,δα,β+1,λ,µ,ρ,p
(
(ω(ax)α); s, r
)
dt. (2.12)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1140 J. C. PRAJAPATI, B. V. NATHWANI
Here
x∫
0
(x− t)η−1 tβ Eγ,δα,β+1,λ,µ,ρ,p
(
(ω(ax)α); s, r
)
dt =
=
x∫
0
(x− t)η−1 tβ
∞∑
n=0
[(γ)δ n]s(ω (at)α)pn+ρ−1
Γ(α(pn+ ρ− 1) + β + 1) [(λ)µ n]r (ρ)pn
dt =
=
∞∑
n=0
[(γ)δ n]s(ω (a)α)pn+ρ−1
Γ(α(pn+ ρ− 1) + β + 1) [(λ)µ n]r (ρ)pn
×
×
x∫
0
(x− t)η−1 tα(pn+ρ−1)+β dt. (2.13)
Take t = xu, then dt = xdu and as t→ 0, u→ 0 and as t→ x, u→ 1
=
∞∑
n=0
[(γ)δn]s (ω aα)pn+ρ−1 xα(pn+ρ−1)+η+β
Γ(α(pn+ ρ− 1) + β) [(λ)µn]r (ρ)pn
1∫
0
(1− u)η−1 uα(pn+ρ−1)+βdt =
=
∞∑
n=0
[(γ)δ n]s (ω aα)pn+ρ−1 xα(pn+ρ−1)+η+β Γ(η) Γ(α(pn+ ρ− 1) + β)
Γ(α(pn+ ρ− 1) + β) [(λ)µ n]r (ρ)pn Γ(α(pn+ ρ− 1) + β + η + 1)
=
= Γ(η)
∞∑
n=0
[(γ)δ n]s (ω (ax)α)pn+ρ−1 x(η+β)
Γ(α(pn+ ρ− 1) + β + η + 1) [(λ)µ n]r (ρ)pn
=
= x(η+β) Γ(η) Eγ,δα,β+η+1,λ,µ,ρ,p
(
ω(ax)α; s, r
)
using this in (2.12) we get (2.11).
Which completes the proof of Theorem 2.6.
Theorem 2.7 (Mellin transform of the operator
(
Eγ,δα,β,λ,µ,ρ,p,ω; 0+f
)
(x)). Let α, β, γ, λ, ρ, ω ∈ C,
Re(α, β, γ, λ, ρ) > 0; δ, µ > 0, p ∈ N, Re(1− S − αρ+ α− β) > 0, then
M
{
(Eγ,δα,β,λ,µ,ρ,p,ω; 0+f)(x);S
}
=
[Γ(λ)]rΓ(ρ) p
2πi [Γ(γ)]sΓ(1− S)
Hr+3,s+1
s+1,r+3×
×
−wtα [(1− γ, δ)]s, (0, p)
(0, 1), (1− S − αρ+ α− β, α p), [(1− λ, µ)]r, (1− ρ, p)
M{tβf(t); S
}
. (2.14)
Proof. By the definition of the Mellin transform, we have
M
{
(Eγ,δα,β,λ,µ,ρ,p,ω; 0+f)(x);S
}
=
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FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1141
=
∞∫
0
xS−1
x∫
0
(x− t)β−1 Eγ,δα,β,λ,µ,ρ,p
(
ω(x− t)α; s, r
)
f(t) dt dx.
Interchanging the order of integration, which is permissible under the given conditions, we find that
M
{
(Eγ,δα,β,λ,µ,ρ,p,ω;0+f)(x);S
}
=
=
∞∫
0
f(t)
∞∫
t
xS−1(x− t)β−1Eγ,δα,β,λ,µ,ρ,p
(
ω(x− t)α; s, r
)
dx dt.
If we set x = t+ u the above integral takes the form
M
{
(Eγ,δα,β,λ,µ,ρ,p,ω;0+f)(x);S
}
=
∞∫
0
f(t)
∞∫
t
(t+ u)S−1uβ−1Eγ,δα,β,λ,µ,ρ,p(ωu
α) du dt.
To evaluate the u-integral, we express the Mittag-Leffler function in terms of its Mellin – Barnes
contour integral by means of the formula (1.14 ), then the above expression transforms into the form
M
{
(Eγ,δα,β,λ,µ,ρ,p,ω;0+f)(x);S
}
=
∞∫
0
f(t)
[Γ(λ)]rΓ(ρ)p(−ω)ρ−1
2πi[Γ(γ)]s
×
×
i∞∫
−i∞
Γ(−pξ) Γ(1 + pξ) [Γ(γ + δξ)]s (−ω)pξ
Γ(β + αρ− α+ αpξ) [Γ(λ+ µξ)]rΓ(ρ+ pξ)
×
×
∞∫
0
(t+ u)s−1uα(pξ+ρ−1)+β−1du dξ dt.
If we evaluate the u-integral with the help of the formula
∞∫
0
xν−1(x+ a)−ρdx =
Γ(ν)Γ(ρ− ν)
Γ(ρ)
, Re(ρ) > Re(ν) > 0,
then after some simplification, it is seen that the right-hand side of above equation simplifies to
[Γ(λ)]rΓ(ρ)p
2πi[Γ(γ)]sΓ(1− s)
i∞∫
−i∞
Γ(−pξ)Γ(1 + pξ)[Γ(γ + δξ)]sΓ(1− s− α(pξ + ρ− 1)− β)
[Γ(λ+ µξ)]rΓ(ρ+ pξ)
×
×(−ωtα)pξ+ρ−1dξ
∞∫
0
tβ+s−1f(t) dt.
By using the definition of H-function yields the desired result.
For s = 1, r = 0, ρ = 1, p = 1, δ = q the Theorem 2.7 reduces to the following corollary.
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1142 J. C. PRAJAPATI, B. V. NATHWANI
Corollary 2.1.
M
{
(Eγ,qα,β,ω;0+f)(x);S
}
=
1
Γ(γ)Γ(1− S)
×
×H2,1
1,2
−wtα (1− γ, q)
(0, 1)(1− S − β, α)
M{tβf(t);S},
where Re(α) > 0, Re(β) > 0, Re(γ) > 0; q ∈ (0, 1) ∪ N, Re(1 − S − β) > 0 and H2,1
1,2 (·) is the
H-function defined by (1.15).
Theorem 2.8 (Laplace transform of the operator
(
Eγ,δα,β,λ,µ,ρ,p,ω;0+f
)
(x)).
L
{
(Eγ,δα,β,λ,µ,ρ,p,ω; 0+f)(x);P
}
=
=
[Γ(λ)]r Γ(ρ) ωρ−1
[Γ(γ)]s P β+αρ−α s+1
ψr+1
[
[(γ, q)]s, (1, 1); ωp/P pα
[(λ, µ)]r, (ρ, p);
]
F (P ),
where Re(α) > 0, Re(β) > 0, Re(γ) > 0; Re(p) > |ω|1/Re(α) and F (P ) is the Laplace transform
of f(t), defined by
L{f(t); p} = F (P ) =
∞∫
0
e−Ptf(t) dt,
where Re(p) > 0 and the integral is convergent.
Proof. By virtue of the definition of Laplace transform, it follows that
L
{
(Eγ,δα,β,λ,µ,ρ,pω; 0+f)(x);P
}
=
∞∫
0
e−Pt
x∫
0
(x− t)β−1Eγ,δα,β,λ,µ,ρ,p[ω(x− t)α]f(t) dt dx.
Interchanging the order of integration, which is permissible under the conditions given in the theorem,
we find that
L
{
(Eγ,δα,β,λ,µ,ρ,p ω; 0+f)(x);P
}
=
∞∫
0
f(t)dt
∞∫
t
e−Pt(x− t)β−1Eγ,δα,β,λ,µ,ρ,p[ω(x− t)α] dx.
If we set x = t+ u we obtain
L
{
(Eγ,δα,β,λ,µ,ρ,pω;0+f)(x);P
}
=
∞∫
0
ePtf(t)dt
∞∫
0
e−Puuβ−1Eγ,δα,β,λ,µ,ρ,p[ωu
α] du.
On making use of the series definition (1.1), the above expression becomes
=
∞∑
k=0
[(γ)δk]
s ω(pk+ρ−1)
Γ(αk + β) [(λ)µk]r (ρ)pk
∞∫
0
ePtf(t)dt
∞∫
0
e−Puuβ+α(pk+ρ−1)−1du =
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FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1143
=
∞∑
k=0
[(γ)δk]
s ω(pk+ρ−1)
P β+α(pk+ρ−1)[(λ)µk]r (ρ)pk
∞∫
0
e−Ptf(t)dt =
=
[Γ(λ)]r Γ(ρ) ωρ−1
[Γ(γ)]s P β+αρ−α
s+1ψr+1
[
[(γ, q)]s, (1, 1); ωp/P pα
[(λ, µ)]r, (ρ, p);
]
F (p)
and F (P ) is the Laplace transforms of f(t).
For s = 1, r = 0, ρ = 1, p = 1, δ = q the Theorem 2.8 reduces to the following corollary.
Corollary 2.2.
L
{
(Eγ,qα,β,ω;0+f)(x);P
}
=
1
Γ(γ)
P−β1 ψ0
[
(γ, q); ω/Pα
−;
]
F (p),
where Re(α) > 0, Re(β) > 0, Re(γ) > 0; Re(P ) > |ω|1/Re(α) and F (P ) is the Laplace transform
of f(t), defined by
L{f(t);P} = F (P ) =
∞∫
0
e−P tf(t) dt,
where Re(P ) > 0 and the integral is convergent.
3. Properties. In this section certain properties of the functions Et(c, ν, γ, δ, λ, µ, ρ, p) and
Et(c,−η, γ, δ, λ, µ, ρ, p) will be obtained. We begin with the function
f(t) =
∞∑
n=0
[(γ)δn]s (ct)(pn+ρ−1)
Γ(ρ) ((ρ)pn)2 [(λ)µn]r
,
where γ ∈ C, δ > 0, c — arbitrary constant.
Now, using (1.4), the fractional integral operator of order ν is given as
Iν f(t) =
1
Γ(ν)
t∫
0
(t− ξ)ν−1
∞∑
n=0
[(γ)δn]s (cξ)(pn+ρ−1)
Γ(ρ) ((ρ)pn)2 [(λ)µn]r
dξ =
=
1
Γ(ν)
∞∑
n=0
[(γ)δn]s c(pn+ρ−1)
Γ(ρ) ((ρ)pn)2 [(λ)µn]r
t∫
0
ξpn+ρ−1(t− ξ)ν−1 dξ.
After some simplification and using (1.1), we can write
Iν f(t) = tν
∞∑
n=0
[(γ)δ n]s (ct)(pn+ρ−1)
Γ(1(pn+ ρ− 1) + ν + 1) ((ρ)pn) [(λ)µ n]r
= (3.1)
= tν Eγ,δ1,ν+1,λ,µ,ρ,p(ct; s, r). (3.2)
We denote the function (3.2) as Et(c, ν, γ, δ, λ, µ, ρ, p), i.e.,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1144 J. C. PRAJAPATI, B. V. NATHWANI
Et(c, ν, γ, δ, λ, µ, ρ, p) = tν Eγ,δ1,ν+1,λ,µ,ρ,p(ct; s, r). (3.3)
Now, using (1.7), the fractional differential operator of order η is given as
Dη f(t) = Dk
[
Ik−η
∞∑
n=0
[(γ)δ n]s (ct)(pn+ρ−1)
Γ(ρ) ((ρ)pn)2
[(λ)µ n]r
]
.
Applying (3.1), after some simplification and using (1.1) it yields
Dη f(t) = t−η
∞∑
n=0
[(γ)δn]s (ct)(pn+ρ−1)
Γ(1(pn+ ρ− 1) + (1− η)) ((ρ)pn) [(λ)µn]r
=
= t−η Eγ,δ1,1−η,λ,µ,ρ,p(ct; s, r). (3.4)
We denote the function (3.4) as Et(c,−η, γ, δ, λ, µ, ρ, p), i.e.,
Et(c,−η, γ, δ, λ, µ, ρ, p) = t−η Eγ,δ1,1−η,λ,µ,ρ,p(ct; s, r). (3.5)
Theorem 3.1. Let γ ∈ C, Re(γ) > 0, δ > 0, c is arbitrary constant and fractional integral and
differential operator is of order σ, then
IσEt(c, ν, γ, δ, λ, µ, ρ, p) = Et(c, σ + ν, γ, δ, λ, µ, ρ, p) (3.6)
and
Dσ
(
Et(c, ν, γ, δ, λ, µ, ρ, p)
)
= Et(c, ν − σ, γ, δ, λ, µ, ρ, p). (3.7)
Proof. From (1.4), we get
IσEt(c, ν, γ, δ, λ, µ, ρ, p) =
1
Γ(σ)
t∫
0
(t− ξ)σ−1 Eξ(c, ν, γ, δ, λ, µ, ρ, p) dξ.
Using (3.3), above equation becomes,
IσEt(c, ν, γ, δ, λ, µ, ρ, p) =
1
Γ(σ)
t∫
0
(t− ξ)σ−1 ξν×
×
∞∑
n=0
[(γ)δ n]s (cξ)(pn+ρ−1)
Γ(1(pn+ ρ− 1) + ν + 1) ((ρ)pn) [(λ)µ n]r
dξ.
Now, substituting ξ = xt, after some simplification and once again use of (3.3) gives (3.6).
From (1.7) and using (3.6), we get
Dσ
(
Et(c, ν, γ, δ, λ, µ, ρ, p)
)
= Dk
{
tk−σ+ν Eγ,δ1,k−η+ν+1,λ,µ,ρ,p(ct; s, r)
}
.
Using (1.1) and (3.3), we get (3.7).
In the light of the Theorem 2.7, we prove following theorem.
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FRACTIONAL CALCULUS OF A UNIFIED MITTAG-LEFFLER FUNCTION 1145
Theorem 3.2. Let η ∈ C, Re(η) > 0, δ > 0, c is arbitrary constant and fractional integral and
differential operator is of order σ, then
IσEt(c,−η, γ, δ, λ, µ, ρ, p) = Et(c, σ − η, γ, δ, λ, µ, ρ, p),
Dσ
(
Et(c,−η, γ, δ, λ, µ, ρ, p)
)
= Et(c,−σ − η, γ, δ, λ, µ, ρ, p).
Acknowledgement. Authors are indebted to Dr. B. I. Dave for his guidance and sincere thanks
to him for his kind help in LaTeX type setting.
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Received 14.05.12
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