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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 nΦkm(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 ρIγ0+ϕ and ρIγ−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 ρDγ0+ϕ and ρDγ−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 + αir), (qj + βjr) ∈ C \ kZ−, the generalized k−Wright function nΦkm isdefined by

nΦ
k
m(z) = nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣z] = ∞∑
r=0

∏n
i=1 Γk(pi + αir)∏m
j=1 Γk(qj + βjr)

zr

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

∣∣−αik 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 + αir), (qj + βjr) ∈C\kZ−

(1) If ∆ > −1, then series (1.12) is absolutely convergent for all z ∈C and generalized k−Wrightfunction nΦkm(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
I
n−γ
0+ τ

α−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
I
n−γ
− τ

α−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
I
n−γ
− e

−λτρ)(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(α
ρ
−γ)

Γ(α
ρ

)
sργ−α. (2.10)



Eur. J. Math. Anal. 1 (2021) 39
3. 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 ρIγ0+ for generalized k−Wright function nΦkm(z)is given as(

ρI
γ
0+

(
τ
α
k
−1

nΦ
k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ νk ]))(s)
= (

k

ρ
)γ s

α
k

+ργ−1
n+1Φ

k
m+1

[ (
pi,αi

)
1,n
,
(

1
ρ

(α + (ρ− 1)k), ν
ρ

)(
qj,βj

)
1,m
,
(

1
ρ

(α + (ρ(γ + 1) − 1)k), ν
ρ

)∣∣∣∣∣ 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

nΦ
k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ νk ]))(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 + αir)∏m
j=1 Γk(qj + βjr)

(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 + αir)∏m
j=1 Γk(qj + βjr)

(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 + αir)∏m
j=1 Γk(qj + βjr)

(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+1Φ

k
m+1

[ (
pi,αi

)
1,n
,
(

1
ρ

(α + (ρ− 1)k), ν
ρ

)(
qj,βj

)
1,m
,
(

1
ρ

(α + (ρ(γ + 1) − 1)k), ν
ρ

)∣∣∣∣∣ 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 ρIγ− for generalized k−Wright function nΦkm(z) isgiven as (

ρI
γ
−

(
τ−

α
k nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ−νk ]))(s)



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

k

ρ
)γ sργ−

α
k n+1Φ

k
m+1

[(
pi,αi

)
1,n
,
(
α
ρ
−kγ, ν

ρ

)(
qj,βj

)
1,m
,
(
α
ρ
, ν
ρ

) ∣∣∣∣∣ λ s−νk
]
. (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 nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ−νk ]))(s).
Using (1.12), we can write the above equation as

N ≡
∞∑
r=0

∏n
i=1 Γk(pi + αir)∏m
j=1 Γk(qj + βjr)

(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 + αir)∏m
j=1 Γk(qj + βjr)

(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+1Φ

k
m+1

[(
pi,αi

)
1,n
,
(
α
ρ
−kγ, ν

ρ

)(
qj,βj

)
1,m
,
(
α
ρ
, ν
ρ

) ∣∣∣∣∣ λ s−νk
]
.

�

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 ρDγ0+ for generalized k−Wright function
nΦ

k
m(z) is given as(

ρD
γ
0+

(
τ
α
k
−1

nΦ
k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ νk ]))(s)
= (

k

ρ
)−γ s

α
k
−ργ−1

n+1Φ
k
m+1

[ (
pi,αi

)
1,n
,
(

1
ρ

(α + (ρ− 1)k), ν
ρ

)(
qj,βj

)
1,m
,
(

1
ρ

(α + (ρ(1 −γ) − 1)k), ν
ρ

)∣∣∣∣∣ 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

nΦ
k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ νk ]))(s).



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

P ≡
(
s1−ρ

d

ds

)n(
ρI
n−γ
0+

(
τ
α
k
−1

nΦ
k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ νk ]))(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+1Φ

k
m+1

[ (
pi,αi

)
1,n
,
(

1
ρ

(α + (ρ− 1)k), ν
ρ

)(
qj,βj

)
1,m
,
(

1
ρ

(α + (ρ(n−γ + 1) − 1)k), ν
ρ

)∣∣∣∣∣ 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 + αir)Γk(

1
ρ

(α + (ρ− 1)k) + ν
ρ
r)∏m

j=1 Γk(qj + βjr)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 + αir)Γk(

1
ρ

(α + (ρ− 1)k) + ν
ρ
r)∏m

j=1 Γk(qj + βjr)Γk(
1
ρ

(α + (ρ(n−γ + 1) − 1)k) + ν
ρ
r)

(λ)r

r!

×
Γ( 1

ρ
(α
k

+ νr
k

+ (n−γ)ρ + ρ− 1)
Γ( 1

ρ
(α
k

+ 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+1Φ
k
m+1

[ (
pi,αi

)
1,n
,
(

1
ρ

(α + (ρ− 1)k), ν
ρ

)(
qj,βj

)
1,m
,
(

1
ρ

(α + (ρ(1 −γ) − 1)k), ν
ρ

)∣∣∣∣∣ 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 ρDγ− for generalized k−Wrightfunction nΦkm(z) is given as(

ρD
γ
−

(
τ−

α
k nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ−νk ]))(s)
= (

k

ρ
)−γ s−ργ−

α
k n+1Φ

k
m+1

[(
pi,αi

)
1,n
,
(
α
ρ

+ kγ, ν
ρ

)(
qj,βj

)
1,m
,
(
α
ρ
, ν
ρ

) ∣∣∣∣∣ λ s−νk
] (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 nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ−νk ]))(s).
Using (1.4), we have

Q ≡
(
− s1−ρ

d

ds

)n(
ρI
n−γ
−

(
τ−

α
k nΦ

k
m

[
(pi,αi )1,n

(qj,βj)1,m

∣∣∣ λ τ−νk ]))(s).



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

Q ≡
(
− s1−ρ

d

ds

)n
(
k

ρ
)n−γ sρ(n−γ)−

α
k n+1Φ

k
m+1

[(
pi,αi

)
1,n
,
(
α
ρ
−k(n−γ), ν

ρ

)(
qj,βj

)
1,m
,
(
α
ρ
, ν
ρ

) ∣∣∣∣∣ λ s−νk
]
.

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

k

ρ
)n−γ

∞∑
r=0

∏n
i=1 Γk(pi + αir)Γk(

α
ρ
− (n−γ)k + ν

ρ
r)∏m

j=1 Γk(qj + βjr)Γk(
α
ρ

+ 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 + αir)Γk(

α
ρ
− (n−γ)k + ν

ρ
r)∏m

j=1 Γk(qj + βjr)Γk(
α
ρ

+ 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 + αir)∏m

j=1 Γk(qj + βjr)Γ(
α
ρk

+ 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+1Φ

k
m+1

[(
pi,αi

)
1,n
,
(
α
ρ

+ kγ, ν
ρ

)(
qj,βj

)
1,m
,
(
α
ρ
, ν
ρ

) ∣∣∣∣∣ λ s−νk
]
.

�



Eur. J. Math. Anal. 1 (2021) 43
5. 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 ρIγ0+ and ρIγ− for generalized k−Wright function
nΦ

k
m(z) are established.

• The Katugampola fractional differentiation ρDγ0+ and ρDγ− for generalized k−Wright func-tion nΦkm(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

