In this article we will presented a new denition of the function pW k;;(z).&nbsp;Some elementary properties of the new pW k;;(z) are presented and his Laplace transform is obtained. Also it has been shown that the fractional Riemann-Liouville integral&nbsp;transform such functions with powers multipliers into functions of the same form with a very interesting relation between indices

Romero, Luis Guillermo

English

PURKH (E-Journals)

The p-k-Wright Function
Ojeda,Nelson R1. Romero,Luis G.2.
1,2Faculty of Humanities
National University of Formosa
Av. Gutnisky 3200. Formosa.3600.ARGENTINA
2e-mail: guille-romero@live.com.ar
Abstract
In this article we will presented a new definition of the function pW
γ
k,α,β(z).
Some elementary properties of the new pW
γ
k,α,β(z) are presented and his Laplace trans-
form is obtained. Also it has been shown that the fractional Riemann-Liouville integral
transform such functions with powers multipliers into functions of the same form with a
very interesting relation between indices
Key Words: p-k-Wrigth function, Laplace Transform, Fractional calculus.
Mathematics Subject Classification: 26A33, 33E12.
I Preliminares
The importance of the role played by the Wright function Wα,β(z) in partial differential equation
of fractional order is well known and was widely treated in papers by several authors including
Gorenflo, Luchko, Mainardi (cf. [5]), Mainardi (cf. [9]), Mainardi, Pagnini (cf. [10]).
Romero - Cerutti (cf. [14]) introduce a generalization of the Wright function denoted by
W γk,α,β(z), that we will named the k-Wright function, in whose definition is used the k-Gamma
function Γk(z) and the k-Pochhammer symbol (γ)n,k defined Diaz and Pariguan (cf. [2]) .
Definition 1. Let k ∈ R+ z, α, β, γ ∈ C , Re(α) > 0,Re(β) > 0.The k-Wrigth function is
defined as
W γk,α,β(z) =
∞∑
n=0
(γ)n,k
Γk(αn+ β)
(z)n
(n!)2
(I.1)
where (γ)n,k is the k-Pochhammer symbol given by
(γ)n,k = γ(γ + k)(γ + 2k) . . . (γ + (n− 1)k)
and Γk(z) is the k-Gamma function Γk(z)
Γk(z) =
∫ ∞
0
e−
tk
k tz−1, z ∈ C ; k ∈ R+,<(z) > 0 (I.2)
1
Recently, K.S. Gehlot (cf.[4]) has introduced a modification of the k-Gamma function by
means of the following integral
pΓk(z) =
∫ ∞
0
e−
tk
p tz−1dt, z ∈ C ; k, p ∈ R+,<(z) > 0 (I.3)
Also, he has define a new Pochhammer symbol
p(z)n,k =
(zp
k
)
.
(zp
k
+ p
)
.
(zp
k
+ 2p
)
...
(zp
k
+ (n− 1)p
)
(I.4)
=
pΓ(z + nk)
pΓ(z)
; z ∈ C,<(z) > 0 (I.5)
where k, p ∈ R+,n ∈ N and also establishes the relation between the function introduced by
him and k-Gamma and the clasical Gamma function . In fact, we have following.
Lemma 1. For the p-k-Gamma function, the k-Gamma function and the classical Gamma
function it is verified:
pΓk(z) =
(p
k
) z
k
Γk(z) =
p
z
k
k
Γ(
z
k
) ; z ∈ C ; k, p ∈ R+,<(z) > 0 (I.6)
And, for the p(z)n,k Pochhammer symbol, we have the following relation
Lemma 2. For the p-k-Pochhammer symbol, the k-Pochhammer symbol and the classical
Pochhammer symbol it has
p(z)n,k =
(p
k
)n
(z)n,k = p
n(z)n ; z ∈ C ; k, p ∈ R+,<(z) > 0 (I.7)
The proof Lemma 1 and Lemma 2 could be seen in (cf.[4]).
In a recent paper (cf. [1]) have introduced a p-k-generalization of the classical Mittag-Leffler
function
pE
γ
k,α,β(z) =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(z)n
n!
(I.8)
where k, p ∈ R+ ,z, α, β, γ ∈ C , Re(α) > 0,Re(β) > 0. As particular cases of pEγk,α,β(z) we
have the classical two parameters Mittag- Leffler function Eα,β(z) for p = k = γ = 1 and for
p = k one gets the k-Mittag- Leffler function Eγk,α,β(z)
In the development of this paper we use fractional integrals and fractional derivatives, and
also Laplace transform, so we introduce the definitions and notations.
Definition 2. Let f be a sufficiently well-behaved function with support in R+ , and let ν
be a real number, ν > 0. The Riemann-Liouville fractional integral of order ν, Iν+f is given by
Iν+f(t)
.
=
1
Γ(ν)
∫ t
0
(t− τ)ν−1f(τ)dτ (I.9)
here Γ(z) denotes the Gamma function of Euler
Γ(z) =
∫ ∞
0
e−ttz−1dt ,<(z) > 0 (I.10)
2
It is known that the semigroup property is verified
Iν+I
µ
+ = I
ν+µ
+ (I.11)
where by I0+ we denote the Identity operator.
The Riemann-Liouville fractional derivative of order ν > 0, Dν+ is defined as the left inverse
of the Riemann-Liouville integral of order ν; i. e,
Dν+I
ν
+ = I , ν > 0
Another way to defined this fractional derivative is as follows.
Definition 3. Let ν > 0 be a real number, and let bem the integer sucht thatm−1 < ν < m
. Then the Riemann-Liouville fractional derivative of order ν is given by
Dν+f(t) = D
mIm−ν+ f(t) (I.12)
Definition 4. Let L(f)(s) be the Laplace transform of an exponential order function and
piecewise continuous where
L [f(t)] (s) =
∫ ∞
0
e−stf(t)dt (I.13)
t ∈ R+ , and s ∈ C.
Exist the relationship between the k-Wrigth function and the k-Mittag-Leffler function
obtained throung the Laplace transform (cf[14]). In fact, we have the following
L[W γk,α,β(z)](s) =
1
s
Eγk,α,β(s
−1) (I.14)
where k ∈ R+ ,α, β, γ ∈ C , <(α) > 0,<(β) > 0 ,<(γ) > 0 and <(s) > 0.
II Main Results
II.1 Definition and convergence
In view of the expressions (I.1),(I.3),(I.8) and (I.14) we introduce a new function of Wright type
that we will call the p-k-Wright function by means of the following
Definition 5. Let k, p ∈ R+ ,α, β, γ ∈ C , Re(α) > 0,Re(β) > 0.The p-k-Wrigth function
is defined as
pW
γ
k,α,β(z) =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(z)n
(n!)2
(II.1)
where pΓk(z) is given by (I.3), and p(γ)n,k is the Pochhammer symbol given by (I.4).
Easily we can prove that pW
γ
k,α,β(z) → W γk,α,β(z) as p → k , because p(γ)n,k → (γ)n,k and
pΓk → Γk.
As particular cases of pW
γ
k,α,β(z) we have the classical two parameters Wright function
Wα,β(z) for p = k = γ = 1.
Theorem 1. The p-k-Wright function, defined in (II.1),is an entire function.
3
Proof. Rewriting the series
pW
γ
k,α,β(z) =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(z)n
(n!)2
=:
∞∑
n=0
cnz
n (II.2)
then radius of convergence of the p-k-Wright function will be called R such that
R = lim
n→∞
sup
∣∣∣∣ cncn+1
∣∣∣∣ (II.3)
Taking into account the proof of Theorem 1. that the p-k-Mittag-Leffler function is an entire
function,the definition of function (II.1) differs in that the factorials is squared ((n!)2), therefore
in an analogous way we can obtein apliying Lemma 1 and (I.5)∣∣∣∣ cncn+1
∣∣∣∣ = (n+ 1)2|pαk−1|
∣∣∣∣∣Γ
(
α
k
n+ α
k
+ β
k
)
Γ
(
α
k
n+ β
k
) ∣∣∣∣∣
∣∣∣∣∣ Γ
(
n+ γ
k
)
Γ
(
n+ γ
k
+ 1
)∣∣∣∣∣ (II.4)
≈ (n+ 1)|pαk−1|
∣∣∣∣(αk n)αk
∣∣∣∣→∞ (II.5)
Thus, the p-k-Wright function is an entire function. 
II.2 Elementary Properties of the p-k-Wright function
In this section we obtain several elementary properties of our p-k-Wright defined by (II.1) and
some others associated with the p-k-function Mittag-Leffler function obtained by means of the
Laplace Transform.
Lemma 3 Let k, p ∈ R+ ,α, β, γ ∈ C , Re(α) > 0,Re(β) > 0. Then there holds the formula
d
dz
[
pW
γ
k,α,β(z)
]
=
(γp
k
) ∞∑
n=0
p(γ + k)n,k
pΓk(αn+ α + β)
zn
(n+ 1)(n!)2
(II.6)
Proof Let α, β, γ ∈ C , Re(α) > 0,Re(β) > 0.From definition (II.1), and Lemma 2 deduced
the well known relations for the p-k-Pochhammer symbol
p(γ)n+1,k =
(γp
k
)
p(γ + k)n,k
then
d
dz
[
pW
γ
k,α,β(z)
]
=
d
dz
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(z)n
(n!)2
=
∞∑
n=1
p(γ)n,k
pΓk(αn+ β)
n
zn−1
(n!)2
=
∞∑
n=0
p(γ)n+1,k
pΓk(α(n+ 1) + β)
(n+ 1)
zn
((n+ 1)!)2
=
(γp
k
) ∞∑
n=0
p(γ + k)n,k
pΓk(αn+ α + β)
zn
(n+ 1)(n!)2
.
4
Theorem 2 Let k, p ∈ R+ ,α, β, γ ∈ C , Re(α) > 0,Re(β) > 0. Then there holds the
formula
pW
γ+k
k,α,β(z)− pW γk,α,β(z) =
(
k
γ
)
z
d
dz
[
pW
γ
k,α,β(z)
]
(II.7)
Proof Let α, β, γ ∈ C , Re(α) > 0,Re(β) > 0.From definition (II.1)
pW
γ+k
k,α,β(z)− pW γk,α,β(z) =
∞∑
n=0
p(γ + k)n,k − (pγ)n,k
pΓk(αn+ β)
(z)n
(n!)2
(II.8)
Taking into account that the Pochhammer p-k-symbol verified p(γ+k)n,k =
[
1 + n
(
k
γ
)]
p(γ)n,k
it result
p(γ + k)n,k − p(γ)n,k = n
(
k
γ
)
p(γ)n,k (II.9)
Replacing (II.9) in (II.8) we have
pW
γ+k
k,α,β(z)− pW γk,α,β(z) =
∞∑
n=0
n
(
k
γ
)
p(γ)n,k
pΓk(αn+ β)
(z)n
(n!)2
=
∞∑
n=1
n
(
k
γ
)
p(γ)n,k
pΓk(αn+ β)
(z)n
(n!)2
=
∞∑
n=0
(n+ 1)
(
k
γ
)
p(γ)n+1,k
pΓk(α(n+ 1) + β)
(z)(n+1)
((n+ 1)!)2
=
(
k
γ
)
z
(γp
k
) ∞∑
n=0
p(γ + k)n,k
pΓk(αn+ α + β)
zn
(n+ 1)(n!)2
Then, by Lemma 3. it results
pW
γ+k
k,α,β(z)− pW γk,α,β(z) =
(
k
γ
)
z
d
dt
[
pW
γ
k,α,β(z)
]
.
Remark Furthermore, it can be proved that verified the following differential equation
pW
γ+k
k,α,β(z) =
(
k
γ
)
z
d
dt
[
pW
γ
k,α,β(z)
]
+ pW
γ
k,α,β(z) (II.10)
Theorem 3 Let k, p ∈ R+ ,α, β, γ ∈ C , Re(α) > 0,Re(β) > 0. Then there holds the
formula
pW
γ
k,α,β(z) =
(αp
k
)
z
d
dz
[
pW
γ
k,α,β+k(z)
]
+
(
βp
k
)
pW
γ
k,α,β+k(z) (II.11)
Proof. By definition (II.1) we have
5
(αp
k
)
z
d
dz
[
pW
γ
k,α,β+k(z)
]
+
(
βp
k
)
pW
γ
k,α,β+k(z) =
(αp
k
)
z
d
dz
∞∑
n=0
p(γ)n,k
pΓk(αn+ β + k)
zn
(n!)2
+
(
βp
k
) ∞∑
n=0
p(γ)n,k
pΓk(αn+ β + k)
zn
(n!)2
=
∞∑
n=0
(
αp
k
)
p(γ)n,k
pΓk(αn+ β + k)
znzn−1
(n!)2
+
∞∑
n=0
(
βp
k
)
p(γ)n,k
pΓk(αn+ β + k)
zn
(n!)2
=
∞∑
n=0
(αn+β)p
k p
(γ)n,k
pΓk(αn+ β + k)
zn
(n!)2
Taking into account pΓ(z + k) =
zp
k
pΓ(z)
(αp
k
)
z
d
dz
[
pW
γ
k,α,β+k(z)
]
+
(
βp
k
)
pW
γ
k,α,β+k(z) =
∞∑
n=0
(αn+β)p
k p
(γ)n,k
(αn+β)p
k p
Γk(αn+ β)
zn
(n!)2
(αp
k
)
z
d
dz
[
pW
γ
k,α,β+k(z)
]
+
(
βp
k
)
pW
γ
k,α,β+k(z) =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
zn
(n!)2
= pW
γ
k,α,β(z)
Then , we have
pW
γ
k,α,β(z) =
(αp
k
)
z
d
dz
[
pW
γ
k,α,β+k(z)
]
+
(
βp
k
)
pW
γ
k,α,β+k(z). (II.12)
In what follows shows the relationship between the p-k-Wrigth function and the p-k-Mittag-
Leffler function obtained throung the Laplace transform. In fact, we have the following
Theorem 1. Let k, p ∈ R+ ,α, β, γ ∈ C , Re(α) > 0,Re(β) > 0 Re(γ) > 0,Re(s) > 0
L[pW
γ
k,α,β(z)](s) =
1
s
pE
γ
k,α,β(s
−1) (II.13)
where pE
γ
k,α,β(z) is the p-k-Mittag-Leffler function given by Cerutti; Luque and Dorrego. (cf.
[1])
Proof. From definition of Laplace Transform and from (II.1) we have
L[pW
γ
k,α,β(z)](s) =
∫ ∞
0
e−sz
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
zn
(n!)2
dz =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
1
(n!)2
∫ ∞
0
e−szzndz (II.14)
Taking into account that the integral in (II.14) is∫ ∞
0
e−szzndz =
Γ(n+ 1)
sn+1
=
n!
sn+1
(II.15)
6
From (II.14) and (II.15) we have
L[pW
γ
k,α,β(z)](s) =
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(s−1)n+1
n!
=
1
s
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
(s−1)n
n!
=
1
s
pE
γ
k,α,β(s
−1)
Then
L[pW
γ
k,α,β(z)](s) =
1
s
pE
γ
k,α,β(s
−1) (II.16)
III Fractional Integral and Fractional Derivative of the
E(t, p, k, α, β) function
In this subsection we will defined the function E(t, p, k, α, β) in term of the p-k- Wrigth function
and then we evaluate its Riemann-Liouville fractional integral and derivative.
Definition 6. Let α, β, γ be complex numbers that <(α) > 0, <(β) > 0 and <(γ) > 0,
k > 0 and t ∈ R. We define the auxiliary function E(t, p, k, α, β) by the following relation
E(t, p, k, α, β) = tβk−1pW γk,α,β(t
α
k ) (III.1)
Easily we can prove that E(t, p, k, α, β) → E(t, k, α, β) as p → k , where E(t, k, α, β) is the
function in term of the k- Wrigth function defined (c.f[14])
Proposition 1.Let α, β, γ be complex numbers that <(α) > 0, <(β) > 0 and <(γ) > 0,
k > 0 and t ∈ R then
Iν+(E(t, p, k, α, β))(x) = pνx
β
k
+ν−1
pW
γ
k,α,β+kν(x
α
k ) (III.2)
Proof. By definition (I.9) and (III.1) we have
Iν+(E(t, p, k, α, β))(x) =
1
Γ(ν)
∫ x
0
t
β
k
−1
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)
t
αn
k
(n!)2
(x− t)ν−1dt
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
∫ x
0
t
αn+β
k
−1(x− t)ν−1dt (III.3)
making the change of variable
t = ξx, x− t = x(1− ξ), dt = xdξ
and replacing in (III.3) it result
Iν+(E(t, p, k, α, β))(x) =
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
∫ 1
0
(ξx)
αn+β
k
−1[x(1− ξ)]ν−1x1dξ =
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
x
αn+β
k
+ν−1
∫ 1
0
ξ
αn+β
k
−1(1− ξ)ν−1dξ = (III.4)
7
The integral in (III.4) result∫ 1
0
ξ
αn+β
k
−1(1− ξ)ν−1dξ = B
(
αn+ β
k
, ν
)
where B(z, w) is the Beta function. Then
Iν+(E(t, p, k, α, β))(x) =
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
x
αn+β
k
+ν−1B
(
αn+ β
k
, ν
)
=
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
x
αn+β
k
+ν−1Γ
(
αn+β
k
)
Γ(ν)
Γ
(
αn+β
k
+ ν
) =
1
Γ(ν)
∞∑
n=0
p(γ)n,k
pΓk(αn+ β)(n!)2
x
αn+β
k
+ν−1
p
αn+β
k
k
pνΓ
(
αn+β
k
)
Γ(ν)
p
αn+β+kν
k
k
Γ
(
αn+β+kν
k
) =
Appliying Lemma 1. , pΓk(z) =
p
z
k
k
Γ( z
k
)
Iν+(E(t, p, k, α, β))(x) =
1
Γ(ν)
x
β
k
+ν−1pν
∞∑
n=0
p(γ)n,k(x
α
k )n
pΓk(αn+ β)(n!)2
pΓk(αn+ β)Γ(ν)
pΓk(αn+ β + kν)
Then,
Iν+(E(t, p, k, α, β))(x) = pνx
β
k
+ν−1
pW
γ
k,α,β+kν(x
α
k )
Proposition 2.Let α, β, γ be complex numbers that <(α) > 0, <(β) > 0 and <(γ) > 0,
k > 0 and t ∈ R then
Dν+(E(t, p, k, α, β))(x) = p−νx
β
k
−ν−1
pW
γ
k,α,β+k(−ν−1)(x
α
k ) (III.5)
Proof. The Riemann-Liouville fractional integral of order r − ν of E(t, p, k, α, β) result
Ir−ν+ (E(t, p, k, α, β))(x) = pr−νx
β
k
+r−ν−1
pW
γ
k,α,β+k(r−ν)(x
α
k ) (III.6)
Now we evaluate the derivative of order r with respect to x of (III.6)
dr
dxr
pr−νx
β
k
+r−ν−1
∞∑
n=0
p(γ)n,k
pΓk(αn+ β + k(r − ν))
x
αn
k
(n!)2
=
dr
dxr
pr−ν
∞∑
n=0
p(γ)n,k
pΓk(αn+ β + k(r − ν))
x
αn+β
k
+r−ν−1
(n!)2
=
pr−ν
∞∑
n=0
p(γ)n,k
(
αn+β
k
+ r − ν − 1) ... (αn+β
k
− ν − 1)
pΓk(αn+ β + k(r − ν))
x
αn+β
k
−ν−1
(n!)2
=
pr−ν
∞∑
n=0
p(γ)n,kΓ
(
αn+β+k(r−ν)
k
− 1
)
p
αn+β+k(r−ν)
k
−1Γ
(
αn+β+k(r−ν)
k
− 1
) xαnk xβk−ν−1
(n!)2Γ
(
αn+β−kν
k
− 1) =
8
pr−ν
∞∑
n=0
p(γ)n,k
prp
αn+β−kν
k
−1Γ
(
αn+β−kν
k
− 1) x
αn
k x
β
k
−ν−1
(n!)2
=
p−νx
β
k
−ν−1
∞∑
n=0
p(γ)n,k
pΓk (αn+ β − kν − k)
x
αn
k
(n!)2
=
then
Dν+(E(t, p, k, α, β))(x) = p−νx
β
k
−ν−1
pW
γ
k,α,β+k(−ν−1)(x
α
k ) .
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10


The p-k-Wright Function

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The p-k-Wright Function

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