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Katugampola Fractional Calculus With Generalized k−Wright Function

Ahmad Y. A. Salamooni1,∗ , D. D. Pawar2

1Department of Mathematics, Faculty of Education Zabid, Hodeidah University, Al-Hodeidah, Yemen
ayousss83@gmail.com

2School of Mathematical Sciences, Swami Ramanand Teerth Marathwada University, Nanded-431606,
India

dypawar@yahoo.com
∗Correspondence: ayousss83@gmail.com

Abstract. In this article, we present some properties of the Katugampola fractional integrals andderivatives. Also, we study the fractional calculus properties involving Katugampola Fractional inte-grals and derivatives of generalized k−Wright function nΦk
m(z).

1. Introduction and Preliminaries
In recent years, researchers have introduced new fractional integral and differential operatorswhich are generalizations of the famous definitions of Riemann-Liouville, Caputo, Hadamard, Hilfer,etc. They have made a qualitative contribution to fractional differential equations. For more details,see [1, 5-7,9-14] and references therein.

Definition 1.1. [9] Let Ω = [a, b], the Katugampola fractional integrals ρI
γ
0+ϕ and ρI

γ
−ϕ of order

γ ∈ C(R(γ) > 0) are defined for ρ > 0, a = 0 and b =∞ as
(ρI

γ
0+ϕ)(s) =

ρ1−γ

Γ(γ)

∫ s

0

τρ−1ϕ(τ)

(sρ − τρ)1−γ dτ (s > 0), (1.1)
and

(ρI
γ
−ϕ)(s) =

ρ1−γ

Γ(γ)

∫ ∞
s

τρ−1ϕ(τ)

(τρ − sρ)1−γ dτ (s > 0), (1.2)
the corresponding Katugampola fractional derivatives ρD

γ
0+ϕ and ρD

γ
−ϕ are defined with (n =

1 + [R(γ)]
) as

(ρD
γ
0+ϕ)(s) :=

(
s1−ρ d

ds

)1+[R(γ)](
ρ
I

1−γ+[R(γ)]
0+ ϕ

)
(s)

Received: 30 Aug 2021.
Key words and phrases. Katugampola fractional integral and derivative; k−Gamma function; k−Wright function.34

https://adac.ee
https://doi.org/10.28924/ada/ma.1.34
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Eur. J. Math. Anal. 1 (2021) 35

=
ργ−[R(γ)]

Γ(1− γ + [R(γ)])

(
s1−ρ d

ds

)1+[R(γ)]
∫ s

0

τρ−1ϕ(τ)

(sρ − τρ)γ−[R(γ)]
dτ (s > 0), (1.3)

and
(ρD

γ
−ϕ)(s) :=

(
− s1−ρ d

ds

)1+[R(γ)](
ρ
I

1−γ+[R(γ)]
− ϕ

)
(s)

=
ργ−[R(γ)]

Γ(1− γ + [R(γ)])

(
− s1−ρ d

ds

)1+[R(γ)]
∫ ∞
s

τρ−1ϕ(τ)

(τρ − sρ)γ−[R(γ)]
dτ (s > 0). (1.4)

Definition 1.2. [2] The generalized K−Gamma function Γk(y) is defined by
Γk(y) = lim

n→∞

n!kn(nk)
y
k
−1

(y)n,k
(k > 0; y ∈ C \ kZ−), (1.5)

where (y)n,k is the k−Pochhammer symbol given as
(y)n,k :=


Γk(y+nk)

Γk(y) (k ∈ R; y ∈ C \ {0})

y(y + k)(y + 2k)...(y + (n − 1)k) (n ∈ N+; y ∈ C)

(1.6)
and for R(y) > 0, the K−Gamma function Γk(y) is defined by the integral

Γk(y) =

∫ ∞
0

xy−1e−
xk

k dx. (1.7)
This gives a relation with Euler’s Gamma function as

Γk(y) = k
y
k
−1Γ(

y

k
). (1.8)

Also, in [8], we have
Γ(1− y)Γ(y) =

π

sin(yπ)
. (1.9)

Definition 1.3. [14] The Beta function B(υ, ω) is defined as
B(υ, ω) =

∫ 1

0

zυ−1(1− z)ω−1dz, R(υ) > 0, R(ω) > 0,

=
Γ(υ)Γ(ω)

Γ(υ + ω)
(1.10)

Furthermore, we have∫ ∞
x̂

(z − x̂)υ−1(z − ŷ)ω−1dz = (x̂ − ŷ)υ+ω−1B(υ, 1− υ − ω),

x̂ > ŷ , 0 < R(υ) < 1−R(ω). (1.11)

Recently, the Generalized K−Wright function introduced by (Gehlot and Prajapati [3]) is definedas follows:



Eur. J. Math. Anal. 1 (2021) 36

Definition 1.4. For k ∈ R+; z ∈ C; pi , qj ∈ C, αi , βj ∈ R (αi , βj 6= 0; i = 1, 2, ..., n; j =

1, 2, ..., m) and (pi + αi r), (qj + βj r) ∈ C \ kZ−, the generalized k−Wright function nΦk
m isdefined by

nΦk
m(z) = nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣z] =

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

z r

r !
, (1.12)

with the convergence conditions described as
∆ =

m∑
j=1

(βj
k

)
−

n∑
i=1

(αi
k

)
;µ =

n∏
i=1

∣∣αi
k

∣∣−αi
k

m∏
j=1

∣∣βj
k

∣∣ βjk ; ν =

m∑
j=1

(qj
k

)
−

n∑
i=1

(pi
k

)
+
n −m

2
.

Lemma 1.1. [3] For k ∈ R+; z ∈ C; pi , qj ∈ C, αi , βj ∈ R (αi , βj 6= 0; i = 1, 2, ..., n; j =

1, 2, ..., m) and (pi + αi r), (qj + βj r) ∈ C \ kZ−

(1) If ∆ > −1, then series (1.12) is absolutely convergent for all z ∈ C and generalized k−Wrightfunction nΦk
m(z) is an entire function of z.

(2) If ∆ = −1, then series (1.12) is absolutely convergent for all |z | < µ and of
|z | = µ,R(µ) >

1

2
.

2. Properties of Katugampola Fractional Integral and Derivative
In this section, we investigate some properties of the Katugampola fractional integrals andderivatives (1.1), (1.2) and (1.3), (1.4) for the power function ϕ(s) = sα−1 and the exponentialfunction e−λ sρ .

Lemma 2.1. Let ρ > 0,R(γ) = 0 and n = 1 + [R(γ)]

(1) If R(α) > 0, then
(ρI

γ
0+τ

α−1)(s) =
ρ−γΓ(1 + α−1

ρ )

Γ(1 + α−1
ρ + γ)

sργ+(α−1) (R(γ) ≥ 0; R(α) > 0) (2.1)
(ρD

γ
0+τ

α−1)(s) =
ργ−nΓ(1 + α−1

ρ )

Γ(1 + α−1
ρ − γ)

s(α−1)−ργ (R(γ) = 0; R(α) > 0). (2.2)
(2) If α ∈ C, then

(ρI
γ
−τ

α−1)(s) =
ρ−γΓ( 1−α

ρ − γ)

Γ( 1−α
ρ )

sργ+(α−1) (R(γ) ≥ 0; R(γ + α) < 1) (2.3)
(ρD

γ
−τ

α−1)(s) =
ργ−nΓ( 1−α

ρ + γ)

Γ( 1−α
ρ )

s(α−1)−ργ (R(γ) = 0; R(γ + α− [R(γ)]) < 1).

(2.4)
(3) If R(λ) > 0, then

(ρI
γ
−e
−λτρ)(s) = (λρ)−γe−λ s

ρ

(R(γ) ≥ 0) (2.5)



Eur. J. Math. Anal. 1 (2021) 37

(ρD
γ
−e
−λτρ)(s) = (λρ)γe−λ s

ρ

(R(γ) = 0). (2.6)
Proof. To prove this Lemma, let the substitution x = τρ

sρ in parts (1) and (2).
(1) Firstly, by the equation (1.1) and the given substitution, we have

(ρI
γ
0+τ

α−1)(s) =
ρ−γsργ+α−1

Γ(γ)

∫ 1

0

x
α−1
ρ

(1− x)1−γ dx

=
ρ−γsργ+α−1

Γ(γ)
B
(
γ, 1 +

α− 1

ρ

)
.

Now, using equation (1.10), we obtain the result (2.1).Secondly, by the equation (1.3), the given substitution and by using the result (2.1), we have
(ρD

γ
0+τ

α−1)(s) =
(
s1−ρ d

ds

)n(
ρ
In−γ0+ τα−1

)
(s)

=
ργ−nΓ(1 + α−1

ρ )

Γ(1 + α−1
ρ + n − γ)

(
s1−ρ d

ds

)n
sρ(n−γ)+α−1

=
ργ−nΓ(1 + α−1

ρ )

Γ(1 + α−1
ρ − γ)

s(α−1)−ργ .

(2) Firstly, by the equation (1.2) and the given substitution, we have
(ρI

γ
−τ

α−1)(s) =
ρ−γsργ+α−1

Γ(γ)

∫ ∞
1

x
α−1
ρ (x − 1)γ−1dx.

Now, using the equation (1.11) with x̂ = 1 and ŷ = 0, we obtain
(ρI

γ
−τ

α−1)(s) =
ρ−γsργ+α−1

Γ(γ)
B
(
γ, 1− γ − (1 +

α− 1

ρ
)
)
.

By using equation (1.10), we obtain the result (2.3).Secondly, by the equation (1.4), the given substitution and by using the result (2.3), we have
(ρD

γ
−τ

α−1)(s) =
(
− s1−ρ d

ds

)n(
ρ
In−γ− τα−1

)
(s)

=
(−1)nργ−nΓ( 1−α

ρ + γ − n)

Γ( 1−α
ρ )

(
s1−ρ d

ds

)n
sρ(n−γ)+α−1

=
(−1)nργ−n

Γ( 1−α
ρ )

Γ( 1−α
ρ + γ − n)Γ(1− [ 1−α

ρ + γ − n])

Γ(1− [γ − α−1
ρ ])

. (2.7)
Also, by using (1.9), we have

Γ(
1− α
ρ

+ γ − n)Γ(1− [
1− α
ρ

+ γ − n]) =
π

sin([ 1−α
ρ + γ − n]π)

=
(−1)nπ

sin([γ − α−1
ρ ]π)

(2.8)
and

1

Γ(1− [γ − α−1
ρ ])

=
Γ(γ − α−1

ρ )

Γ(γ − α−1
ρ )Γ(1− [γ − α−1

ρ ])
=

Γ(γ − α−1
ρ )

π
sin([γ −

α− 1

ρ
]π) (2.9)

Substituting relations (2.8) and (2.9) in (2.7), we obtain (2.4).



Eur. J. Math. Anal. 1 (2021) 38

(3) For this part, let the substitution x = τρ − sρ.Firstly, by the equation (1.2) and the given substitution in this part, we have
(ρI

γ
−e
−λτρ)(s) =

ρ−γ

Γ(γ)
e−λ s

ρ

∫ ∞
0

e−λ xxγ−1dx,

then by use the substitution ϑ = λ x, we obtain
(ρI

γ
−e
−λτρ)(s) =

ρ−γ

Γ(γ)
e−λ s

ρ

λ−γ
∫ ∞

0

e−ϑϑγ−1dϑ,

since ∫∞
0 e−ϑϑγ−1dϑ = Γ(γ) [8], then the result is satisfied.Secondly, by the equation (1.4) and by using the result (2.5), we have

(ρD
γ
−e
−λτρ)(s) =

(
− s1−ρ d

ds

)n(
ρ
In−γ− e−λτ

ρ)
(s)

= (−1)n
(
s1−ρ d

ds

)n(
(λρ)γ−ne−λ s

ρ)
= (−1)n s(1−ρ)n (λρ)γ−n

( dn
dsn

e−λ s
ρ)

= (λρ)γe−λ s
ρ

.

�

Remark 2.1. (a) In Lemma 2.1, if the power function is ϕ(s) =
(
sρ

ρ

)α−1
, then

(1) If R(α) > 0, then(
ρI
γ
0+

(τρ
ρ

)α−1
)

(s) =
Γ(α)

Γ(α+ γ)

(sρ
ρ

)α+γ−1
(R(γ) ≥ 0; R(α) > 0)

(
ρD

γ
0+

(τρ
ρ

)α−1
)

(s) =
Γ(α)

Γ(α− γ)

(sρ
ρ

)α−γ−1
(R(γ) = 0; R(α) > 0).

(2) If α ∈ C, then(
ρI
γ
−
(τρ
ρ

)α−1
)

(s) =
Γ(1− γ − α)

Γ(1− α)

(sρ
ρ

)α+γ−1
(R(γ) ≥ 0; R(γ + α) < 1)

(
ρD

γ
−
(τρ
ρ

)α−1
)

(s) =
Γ(1 + γ − α)

Γ(1− α)

(sρ
ρ

)α−γ−1
(R(γ) = 0; R(γ + α− [R(γ)]) < 1).

(b) If R(α) > R(γ) > 0, then
(ρI

γ
−τ
−α)(s) =

ρ−γΓ(αρ − γ)

Γ(αρ )
sργ−α. (2.10)



Eur. J. Math. Anal. 1 (2021) 393. Katugampola Fractional integration for Generalized k−Wright Function
In this section, we establish the Katugampola fractional integration for generalized k−Wrightfunction (1.12).

Theorem 3.1. Let γ, α ∈ C such that R(γ) > 0, R(α) > 0; λ ∈ C, ρ > 0, ν > 0, then for
∆ > −1, the Katugampola fractional integration ρI

γ
0+ for generalized k−Wright function nΦk

m(z)is given as(
ρI
γ
0+

(
τ
α
k
−1

nΦk
m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ ν
k

]))
(s)

= (
k

ρ
)γ s

α
k

+ργ−1
n+1Φk

m+1

[ (
pi , αi

)
1,n
,
(

1
ρ(α+ (ρ− 1)k), νρ

)(
qj , βj

)
1,m
,
(

1
ρ(α+ (ρ(γ + 1)− 1)k), νρ

)∣∣∣∣∣ λ s νk
]
. (3.1)

Proof. According to Lemma 1.1, a generalized k−Wright function in both sides of the equation (3.1)exists for s > 0. We consider that
M ≡

(
ρI
γ
0+

(
τ
α
k
−1

nΦk
m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ ν
k

]))
(s).

Using (1.12), we can write the above equation as
M ≡

(
ρI
γ
0+

(
τ
α
k
−1

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

(λ τ
ν
k )r

r !

))
(s).

Now, using the integration of the series term by term, we obtain
M ≡

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

(λ)r

r !

(
ρI
γ
0+

(
τ
α
k

+ νr
k
−1
))

(s).

Applying (2.1), the above equation is reduced to
M ≡

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

(λ)r

r !

ρ−γΓ(1 +
α
k

+ νr
k
−1

ρ )

Γ(1 +
α
k

+ νr
k
−1

ρ + γ)
s
α+νr
k

+ργ−1.

Using (1.8), we obtain
M ≡ (

k

ρ
)γ s

α
k

+ργ−1
n+1Φk

m+1

[ (
pi , αi

)
1,n
,
(

1
ρ(α+ (ρ− 1)k), νρ

)(
qj , βj

)
1,m
,
(

1
ρ(α+ (ρ(γ + 1)− 1)k), νρ

)∣∣∣∣∣ λ s νk
]
.

�

Theorem 3.2. Let γ, α ∈ C such that R(γ) > 0, R(α) > 0; λ ∈ C, ρ > 0, ν > 0, then for
∆ > −1, the Katugampola fractional integration ρI

γ
− for generalized k−Wright function nΦk

m(z) isgiven as (
ρI
γ
−

(
τ−

α
k nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ− ν
k

]))
(s)



Eur. J. Math. Anal. 1 (2021) 40

= (
k

ρ
)γ sργ−

α
k n+1Φk

m+1

[(
pi , αi

)
1,n
,
(
α
ρ − kγ,

ν
ρ

)(
qj , βj

)
1,m
,
(
α
ρ ,

ν
ρ

) ∣∣∣∣∣ λ s− ν
k

]
. (3.2)

Proof. According to Lemma 1.1, a generalized k−Wright function in both sides of the equation (3.2)exists for s > 0. We consider that
N ≡

(
ρI
γ
−

(
τ−

α
k nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ− ν
k

]))
(s).

Using (1.12), we can write the above equation as
N ≡

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

(λ)r

r !

(
ρI
γ
−

(
τ−

α+νr
k

))
(s).

Applying (2.10), the above equation is reduced to
N ≡

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m
j=1 Γk(qj + βj r)

(λ)r

r !

ρ−γΓ(
α+νr
k
ρ − γ)

Γ(
α+νr
k
ρ )

sργ−
α+νr
k .

Using (1.8), we obtain
N ≡ (

k

ρ
)γ sργ−

α
k n+1Φk

m+1

[(
pi , αi

)
1,n
,
(
α
ρ − kγ,

ν
ρ

)(
qj , βj

)
1,m
,
(
α
ρ ,

ν
ρ

) ∣∣∣∣∣ λ s− ν
k

]
.

�

4. Katugampola Fractional Differentiation for Generalized k−Wright Function
This section deals with the Katugampola fractional differentiation for generalized k−Wrightfunction (1.12).

Theorem 4.1. Let γ, α ∈ C such that R(γ) > 0, R(α) > 0; λ ∈ C, ρ > 0, ν > 0, thenfor ∆ > −1, the Katugampola fractional differentiation ρD
γ
0+ for generalized k−Wright function

nΦk
m(z) is given as(

ρD
γ
0+

(
τ
α
k
−1

nΦk
m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ ν
k

]))
(s)

= (
k

ρ
)−γ s

α
k
−ργ−1

n+1Φk
m+1

[ (
pi , αi

)
1,n
,
(

1
ρ(α+ (ρ− 1)k), νρ

)(
qj , βj

)
1,m
,
(

1
ρ(α+ (ρ(1− γ)− 1)k), νρ

)∣∣∣∣∣ λ s νk
]
. (4.1)

Proof. According to Lemma 1.1, a generalized k−Wright function in both sides of the equation (4.1)exists for s > 0. Let n = 1 + [R(γ)]. Then, we consider that
P ≡

(
ρD

γ
0+

(
τ
α
k
−1

nΦk
m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ ν
k

]))
(s).



Eur. J. Math. Anal. 1 (2021) 41Using (1.3), we have
P ≡

(
s1−ρ d

ds

)n(
ρI
n−γ
0+

(
τ
α
k
−1

nΦk
m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ ν
k

]))
(s).

Using Theorem 3.1, we obtain
P ≡

(
s1−ρ d

ds

)n(
(
k

ρ
)n−γ s

α
k

+ρ(n−γ)−1
n+1Φk

m+1

[ (
pi , αi

)
1,n
,
(

1
ρ (α+ (ρ− 1)k), νρ

)(
qj , βj

)
1,m
,
(

1
ρ (α+ (ρ(n − γ + 1)− 1)k), νρ

)∣∣∣∣∣ λ s νk
])
.

Using (1.12), we can write the above equation as
P ≡ (

k

ρ
)n−γ

∞∑
r=0

∏n
i=1 Γk(pi + αi r)Γk( 1

ρ (α+ (ρ− 1)k) + ν
ρ r)∏m

j=1 Γk(qj + βj r)Γk( 1
ρ (α+ (ρ(n − γ + 1)− 1)k) + ν

ρ r)

(λ)r

r !

(
s1−ρ d

ds

)n(
s
α
k

+ ν
k

+ρ(n−γ)−1
)
.

Also, the above equation can be written as
P ≡ kn−γ ργ

∞∑
r=0

∏n
i=1 Γk(pi + αi r)Γk( 1

ρ(α+ (ρ− 1)k) + ν
ρ r)∏m

j=1 Γk(qj + βj r)Γk( 1
ρ(α+ (ρ(n − γ + 1)− 1)k) + ν

ρ r)

(λ)r

r !

×
Γ( 1

ρ(αk + νr
k + (n − γ)ρ+ ρ− 1)

Γ( 1
ρ(αk + νr

k − γρ+ ρ− 1)
s
α
k

+ ν
k
−ργ−1.

Using (1.8), we obtain
P ≡ (

k

ρ
)−γ s

α
k
−ργ−1

n+1Φk
m+1

[ (
pi , αi

)
1,n
,
(

1
ρ(α+ (ρ− 1)k), νρ

)(
qj , βj

)
1,m
,
(

1
ρ(α+ (ρ(1− γ)− 1)k), νρ

)∣∣∣∣∣ λ s νk
]
.

�

Theorem 4.2. Let γ, α ∈ C such that R(γ) > 0, R(α) > 1+[R(γ)]−R(γ); λ ∈ C, ρ > 0, ν >

0, then for ∆ > −1, the Katugampola fractional differentiation ρD
γ
− for generalized k−Wrightfunction nΦk

m(z) is given as(
ρD

γ
−

(
τ−

α
k nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ− ν
k

]))
(s)

= (
k

ρ
)−γ s−ργ−

α
k n+1Φk

m+1

[(
pi , αi

)
1,n
,
(
α
ρ + kγ, νρ

)(
qj , βj

)
1,m
,
(
α
ρ ,

ν
ρ

) ∣∣∣∣∣ λ s− ν
k

] (4.2)
Proof. According to Lemma 1.1, a generalized k−Wright function in both sides of the equation (4.2)exists for s > 0. Let n = 1 + [R(γ)]. Then, we consider that

Q ≡

(
ρD

γ
−

(
τ−

α
k nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ− ν
k

]))
(s).

Using (1.4), we have
Q ≡

(
− s1−ρ d

ds

)n(
ρI
n−γ
−

(
τ−

α
k nΦk

m

[
(pi , αi)1,n

(qj , βj)1,m

∣∣∣ λ τ− ν
k

]))
(s).



Eur. J. Math. Anal. 1 (2021) 42Using Theorem 3.2, we obtain
Q ≡

(
− s1−ρ d

ds

)n
(
k

ρ
)n−γ sρ(n−γ)−α

k n+1Φk
m+1

[(
pi , αi

)
1,n
,
(
α
ρ − k(n − γ), νρ

)(
qj , βj

)
1,m
,
(
α
ρ ,

ν
ρ

) ∣∣∣∣∣ λ s− ν
k

]
.

Using (1.12), we can write the above equation as
Q ≡ (−1)n(

k

ρ
)n−γ

∞∑
r=0

∏n
i=1 Γk(pi + αi r)Γk(αρ − (n − γ)k + ν

ρ r)∏m
j=1 Γk(qj + βj r)Γk(αρ + ν

ρ r)

(λ)r

r !

(
s1−ρ d

ds

)n(
sρ(n−γ)−α

k
− ν
k

)
.

On simplifying the above equation, we obtain
Q ≡ (−1)nkn−γργ

∞∑
r=0

∏n
i=1 Γk(pi + αi r)Γk(αρ − (n − γ)k + ν

ρ r)∏m
j=1 Γk(qj + βj r)Γk(αρ + ν

ρ r)

(λ)r

r !

×
Γ(1 + (n − γ)− α

ρk −
ν
ρk r)

Γ(1− γ − α
ρk −

ν
ρk r)

(
s−ργ−

α
k
− ν
k

)
.

Using (1.8), we obtain
Q ≡ (−1)nργ

∞∑
r=0

∏n
i=1 Γk(pi + αi r)∏m

j=1 Γk(qj + βj r)Γ( αρk + ν
ρk r)

(λ)r

r !

×
Γ(γ − n + α

ρk + ν
ρk r)Γ(1− (γ − n + α

ρk + ν
ρk r))

Γ(1− (γ + α
ρk + ν

ρk r))

(
s−ργ−

α
k
− ν
k

)
. (4.3)

Using (1.9), we have
Γ(γ − n +

α

ρk
+
ν

ρk
r)Γ(1− (γ − n +

α

ρk
+
ν

ρk
r))

=
π

sin[(γ + α
ρk + ν

ρk r)π − nπ]

=
π

sin[(γ + α
ρk + ν

ρk r)π] cos(nπ)

=
(−1)nπ

sin[(γ + α
ρk + ν

ρk r)π]
(4.4)

and
1

Γ(1− (γ + α
ρk + ν

ρk r))
=

Γ(γ + α
ρk + ν

ρk r) sin[(γ + α
ρk + ν

ρk r)π]

π
. (4.5)

Substituting (4.4) and (4.5) in (4.3) and finally by using (1.8), we obtain
Q ≡ (

k

ρ
)−γ s−ργ−

α
k n+1Φk

m+1

[(
pi , αi

)
1,n
,
(
α
ρ + kγ, νρ

)(
qj , βj

)
1,m
,
(
α
ρ ,

ν
ρ

) ∣∣∣∣∣ λ s− ν
k

]
.

�



Eur. J. Math. Anal. 1 (2021) 435. Concluding Remarks
• If ρ = 1, thenTheorems 3.1, 3.2, 4.1 and 4.2, are reduced to Theorems 2, 3, 4 and 5 respectively(see [4]).
• Some general properties of the Katugampola fractional integrals and derivatives for thepower function ϕ(s) = sα−1 and the exponential function e−λ sρ are investigated.
• The Katugampola fractional integration ρI

γ
0+ and ρI

γ
− for generalized k−Wright function

nΦk
m(z) are established.

• The Katugampola fractional differentiation ρD
γ
0+ and ρD

γ
− for generalized k−Wright func-tion nΦk

m(z) are established.
Acknowledgment

The authors are would like to thank the reviewers for their important remarks and suggestions.
References

[1] R. Almeida, A.B. Malinowska, T. Odzijewicz, Fractional differential equations with dependence on the Ca-puto?Katugampola derivative, J. Comput. Nonlinear Dynam. 11 (2016) 061017. https://doi.org/10.1115/1.
4034432.[2] R. Diaz, E. Pariguan, On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Math. 15 (2) (2007)179-192.[3] K.S. Gehlot, J.C. Prajapati, On generalization of K−Wright function and its properties, Pac. J. Appl. Math. 5 (2)(2013) 81-88.[4] K.S. Gehlot, J.C. Prajapati, Fractional calculus of generalized K−Wright function, J. Fraction. Calc. Appl. 4 (2)(2013) 83-289.[5] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011) 860-865.
https://doi.org/10.1016/j.amc.2011.03.062.[6] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (4), (2014), 1-15.[7] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations,arXiv:1411.5229v2[math.CA] 9 Jun (2014).[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier,Amsterdam (2006).[9] D.S. Oliveira, E.C. De Oliveira, Hilfer-Katugampola fractional derivative, Comput. Appl. Math. 37 (2018), 3672-3690.
https://doi.org/10.1007/s40314-017-0536-8.[10] A.Y.A. Salamooni, D.D. Pawar, Unique positive solution for nonlinear Caputo-type fractional q-difference equationswith nonlocal and Stieltjes integral boundary conditions, Fraction. Differ. Calc. 9 (2) (2019), 295-307.[11] A.Y.A. Salamooni, D.D. Pawar, Existence and uniqueness of generalised fractional Cauchy-type problem, Univ. J.Math. Appl. 3 (3) (2020), 121-128.[12] A.Y.A. Salamooni, D.D. Pawar, Existence and uniqueness of boundary value problems for Hilfer-Hadamard-typefractional differential equations, Ganita, 70 (2) (2020), 01-16.

https://doi.org/10.1115/1.4034432
https://doi.org/10.1115/1.4034432
https://doi.org/10.1016/j.amc.2011.03.062
https://doi.org/10.1007/s40314-017-0536-8


Eur. J. Math. Anal. 1 (2021) 44

[13] A.Y.A. Salamooni, D.D. Pawar, Existence and stability results for Hilfer-Katugampola-type fractional implicit dif-ferential equations with nonlocal conditions, J. Nonlinear Sci. Appl. 14 (3) (2021), 124-138. http://dx.doi.org/
10.22436/jnsa.014.03.02.[14] A.Y.A. Salamooni, D.D. Pawar, Existence and uniqueness of nonlocal boundary conditions for Hilfer-Hadamard-type fractional differential equations, Adv. Differ. Equations, 2021 (2021), 198. https://doi.org/10.1186/

s13662-021-03358-0.[15] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon andBreach, New York (1993).

http://dx.doi.org/10.22436/jnsa.014.03.02
http://dx.doi.org/10.22436/jnsa.014.03.02
https://doi.org/10.1186/s13662-021-03358-0
https://doi.org/10.1186/s13662-021-03358-0

	1. Introduction and Preliminaries
	2. Properties of Katugampola Fractional Integral and Derivative
	3. Katugampola Fractional integration for Generalized k-Wright Function
	4. Katugampola Fractional Differentiation for Generalized k-Wright Function
	5. Concluding Remarks
	Acknowledgment
	References

