An extension of Caputo fractional derivative operator and its applications

Research Article

An extension of Caputo fractional derivative operator and its applications

˙I. Onur Kıymaz^a, Ay¸seg¨ul C¸ etinkaya^a,∗, Praveen Agarwal^b

aAhi Evran Univ., Dept. of Mathematics, 40100 Kır¸sehir, Turkey.

bAnand International College of Eng., Dept. of Mathematics, 303012 Jaipur, India.

Communicated by X.-J. Yang

Abstract

In this paper, an extension of Caputo fractional derivative operator is introduced, and the extended fractional derivatives of some elementary functions are calculated. At the same time, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are ob- tained, and Mellin transforms of some extended fractional derivatives are also determined. c2016 All rights reserved.

Keywords: Caputo fractional derivative, hypergeometric functions, generating functions, Mellin transform, integral representations.

2010 MSC: 26A33, 33C05, 33C20, 33C65.

1. Introduction

Special functions are used in the application of mathematics to physical and engineering problems.

In recent years, many authors considered the several extensions of well known special functions (see, for example, [2, 3, 9, 12, 13]; see also the very recent work [8, 10]). In 1994, Chaudhry and Zubair [4], introduced the generalized representation of gamma function. In 1997, Chaudhry et al. [2] presented the following extension of Euler’s beta function

Bp(x, y) :=

Z 1 0

t^x−1(1−t)^y−1e

−p

t(1−t)

dt,

∗Corresponding author

Email addresses: [email protected](˙I. Onur Kıymaz),[email protected](Ay¸seg¨ul C¸ etinkaya), [email protected](Praveen Agarwal)

Received 2016-02-22

where Re(p)>0, Re(x)>0, Re(y)>0.

Recently, Chaudhry et al. [3] usedBp(x, y) to extend the hypergeometric functions as F_p(α, β;γ;z) :=

∞

X

n=0

(α)_n n!

Bp(β+n, γ−β) B(β, γ−β) zⁿ,

wherep≥0,Re(γ)> Re(β)>0 and|z|<1. The symbol (α)_n denotes the Pochhammer’s symbol defined by

(α)_n:= Γ(a+n)

Γ(a) , (α)₀ := 1.

Afterwards, in [1] ¨Ozarslan and ¨Ozergin obtained linear and bilinear generating relations for extended hypergeometric functions by defining the extension of the Riemann-Liouville fractional derivative operator as

D^α,p_z f(z) := d^m

dz^mD^α−m_z f(z)

= d^m dz^m

( 1 Γ(−α+m)

Z z

0

(z−t)^−α+m−1e

−pz2 t(z−t)

f(t)dt )

,

whereRe(p)>0 and m−1< Re(α)< m, m∈N. It is obvious that, these extensions given above, coincide with original ones whenp= 0.

The above-mentioned works have largely motivated our present study. The principle aim of the paper is to present extension of the Caputo fractional derivative operator and calculating the extended fractional derivatives of some elementary functions. In the sequel, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are obtained, and Mellin transforms of some extended fractional derivatives are also determined.

2. Extended hypergeometric functions

In this section, we introduce the extensions of Gauss hypergeometric function₂F₁, the Appell hypergeo- metric functionsF1, F2 and the Lauricella hypergeometric functionF_D,p³ . Throughout this paper we assume thatRe(p)>0 and m∈N.

Definition 2.1. The extended Gauss hypergeometric function is

2F₁(a, b;c;z;p) :=

∞

X

n=0

(a)_n(b)_n (b−m)_n

B_p(b−m+n, c−b+m) B(b−m, c−b+m)

zⁿ

n! (2.1)

for all |z|<1 where m < Re(b)< Re(c).

Definition 2.2. The extended Appell hypergeometric functionF1 is F1(a, b, c;d;x, y;p) :=

∞

X

n,k=0

(a)n+k(b)n(c)k

(a−m)_n+k

Bp(a−m+n+k, d−a+m) B(a−m, d−a+m)

xⁿ n!

y^k

k! (2.2)

for all |x|<1,|y|<1 where m < Re(a)< Re(d).

Definition 2.3. The extended Appell hypergeometric functionF₂ is

F₂(a, b, c;d, e;x, y;p) :=

∞

X

n,k=0

(a)n+k(b)n(c)k

(b−m)n(e)_k

B_p(b−m+n, d−b+m) B(b−m, d−b+m)

xⁿ n!

y^k

k! (2.3)

=

∞

X

n,k=0

(a)n+k(b)n(c)k

(d)_n(c−m)_k

Bp(c−m+k, e−c+m) B(c−m, e−c+m)

xⁿ n!

y^k

k! (2.4)

=

∞

X

n,k=0

"

(a)_n+k(b)_n(c)_k (b−m)n(c−m)k

B_p(b−m+n, d−b+m) B(b−m, d−b+m)

B_p(c−m+k, e−c+m) B(c−m, e−c+m)

xⁿy^k n!k!

#

(2.5) for all |x|+|y|<1 where m < Re(b)< Re(d) and m < Re(c)< Re(e).

Definition 2.4. The extended Lauricella hypergeometric functionF_D,p³ is F_D,p³ (a, b, c, d;e;x, y, z;p) :=

∞

X

n,k,r=0

(a)n+k+r(b)n(c)k(d)r

(a−m)_n+k+r

Bp(a−m+n+k+r, e−a+m) B(a−m, e−a+m)

xⁿ n!

y^k k!

z^r

r! (2.6) for all p

|x|+p

|y|+p

|z|<1 where m < Re(a)< Re(e).

Note that whenp= 0, these functions reduces to well known Gauss hypergeometric function2F1, Appell functionsF₁,F₂ and Lauricella function F_D³, respectively.

3. Extended Caputo fractional derivative operator

The fractional derivative operators has gained considerable popularity and importance during the past few years. In literature point of view many fractional derivative operators already proved their importance.

Very recently, fractional operator, whose derivative has singular kernel introduced by Yang et al. [14].

Motivated by above work many researchers applied new derivative in certain real world problems (see, e.g., [5, 7, 15–17]). In the sequel, we aim to extend the definition of the classical Caputo fractional derivative operator.

The classical Caputo fractional derivative is defined by D^µf(z) := 1

Γ(m−µ) Z z

0

(z−t)^m−µ−1 d^m

dt^mf(t)dt,

wherem−1< Re(µ)< m, m∈N. We refer [6] to the reader for more information about fractional calculus.

Inspired by the same idea in [1], we introduce theExtended Caputo Fractional Derivative as D^µ,p_z f(z) := 1

Γ(m−µ) Z z

0

(z−t)^m−µ−1e

−pz2 t(z−t)

d^m

dt^mf(t)dt, (3.1)

whereRe(p)>0 andm−1< Re(µ)< m, m∈N. In the casep= 0, extended Caputo fractional derivative reduces to classical Caputo Fractional derivative, and also when µ=m∈N0 and p= 0,

D^m,0_z f(z) :=f^(m)(z).

Now, we begin our investigation by calculating the extended fractional derivatives of some elementary functions.

Theorem 3.1. Let m−1< Re(µ)< m and Re(µ)< Re(λ) then D_z^µ,pn

z^λo

= Γ(λ+ 1)B_p(λ−m+ 1, m−µ)

Γ(λ−µ+ 1)B(λ−m+ 1, m−µ)z^λ−µ.

Proof. With direct calculation, we get D^µ,p_z n

z^λo

= 1

Γ(m−µ) Z z

0

(z−t)^m−µ−1e

−pz2 t(z−t)

d^m dt^mt^λdt

= 1

Γ(m−µ)

Γ(λ+ 1) Γ(λ−m+ 1)

Z z

0

(z−t)^m−µ−1t^λ−me

−pz2 t(z−t)

dt

= z^λ−µ Γ(m−µ)

Γ(λ+ 1) Γ(λ−m+ 1)

Z 1 0

(1−u)^m−µ−1u^λ−me

_−p

u(1−u)

du

= Γ(λ+ 1)B_p(λ−m+ 1, m−µ)

Γ(λ−µ+ 1)B(λ−m+ 1, m−µ)z^λ−µ. Remark 3.2. Note that,D^µ,pz

z^λ = 0 for λ= 0,1, . . . , m−1.

The next theorem expresses the extended Caputo fractional derivative of an analytic function.

Theorem 3.3. If f(z) is an analytic function on the disk | z |< ρ and has a power series expansion f(z) =P∞

n=0anzⁿ, then

D^µ,p_z {f(z)}=

∞

X

n=0

a_nD^µ,p_z {zⁿ}

where m−1< Re(µ)< m.

Proof. Using the power series expansion of f, we get D_z^µ,p

f(z)

= 1

Γ(m−µ) Z z

0

(z−t)^m−µ−1e

−pz2 t(z−t)

∞

X

n=0

an

d^m dt^mtⁿdt.

Since the power series converges uniformly and the integral converges absolutely, then the order of the integration and the summation can be changed. So we get,

D_z^µ,p

f(z)

=

∞

X

n=0

an

1 Γ(m−µ)

Z z

0

(z−t)^m−µ−1e

−pz2 t(z−t)

d^m dt^mtⁿdt

!

=

∞

X

n=0

a_nD_z^µ,p{zⁿ}.

The proof of the following theorem is obvious from Theorem 3.1 and 3.3.

Theorem 3.4. If f(z) is an analytic function on the disk | z |< ρ and has a power series expansion f(z) =P∞

n=0a_nzⁿ, then

D_z^µ,p n

z^λ−1f(z) o

=

∞

X

n=0

anD_z^µ,p n

z^λ+n−1 o

=Γ(λ)z^λ−µ−1 Γ(λ−µ)

∞

X

n=0

a_n (λ)_n (λ−µ)_n

Bp(λ−m+n, m−µ) B(λ−m+n, m−µ)zⁿ

=Γ(λ)z^λ−µ−1 Γ(λ−µ)

∞

X

n=0

a_n (λ)_n (λ−m)n

B_p(λ−m+n, m−µ) B(λ−m, m−µ) zⁿ, where m−1< Re(µ)< m < Re(λ).

The following theorems will be useful for finding the generating function relations.

Theorem 3.5. Let m−1< Re(λ−µ)< m < Re(λ), then D_z^λ−µ,p

z^λ−1(1−z)^−α

= Γ(λ)z^µ−1 Γ(µ)

∞

X

n=0

(α)_n(λ)_n (λ−m)_n

B_p(λ−m+n, µ−λ+m) B(λ−m, µ−λ+m)

zⁿ n!

= Γ(λ)

Γ(µ)z^µ−12F1(α, λ;µ;z;p)

(3.2)

for |z|<1.

Proof. If we use the power series expansion of (1−z)^−α and (2.1), we get D_z^λ−µ,p{z^λ−1(1−z)^−α}=D^λ−µ,p_z

( z^λ−1

∞

X

n=0

(α)_nzⁿ n!

)

=

∞

X

n=0

(α)n

n! D^λ−µ,p_z n

z^λ+n−1 o

=

∞

X

n=0

(α)n

n!

Γ(λ+n) Γ(µ+n)

Bp(λ−m+n, m−λ+µ)

B(λ−m+n, m−λ+µ)z^µ+n−1

= Γ(λ) Γ(µ)z^µ−1

∞

X

n=0

(α)_n(λ)_n (µ)n

B_p(λ−m+n, m−λ+µ) B(λ−m+n, m−λ+µ)

zⁿ n!

= Γ(λ) Γ(µ)z^µ−1

∞

X

n=0

(α)n(λ)n

(λ−m)n

B_p(λ−m+n, µ−λ+m) B(λ−m, µ−λ+m)

zⁿ n!

= Γ(λ)

Γ(µ)z^µ−1₂F₁(α, λ;µ;z;p).

Theorem 3.6. Let m−1< Re(λ−µ)< m < Re(λ), then D_z^λ−µ,p

z^λ−1(1−az)^−α(1−bz)^−β

= Γ(λ) Γ(µ)z^µ−1

∞

X

n,k=0

(λ)_n+k(α)_n(β)_k (λ−m)n+k

B_p(λ−m+n+k, µ−λ+m) B(λ−m, µ−λ+m)

(az)ⁿ n!

(bz)^k k!

= Γ(λ)

Γ(µ)z^µ−1F1(λ, α, β;µ;az;bz;p)

(3.3)

for |az |<1 and |bz|<1.

Proof. Using the power series expansion of (1−az)^−α, (1−bz)^−β and (2.2), we get D_z^λ−µ,p

z^λ−1(1−az)^−α(1−bz)^−β

=D_z^λ−µ,p ( _∞

X

n=0

∞

X

k=0

(α)n

n!

(β)k

k! aⁿb^kz^λ+n+k−1 )

=

∞

X

n,k=0

(α)_n n!

(β)_k

k! aⁿb^kD^λ−µ,p_z n

z^λ+n+k−1o

=

∞

X

n,k=0

(α)n

n!

(β)k

k! aⁿb^kΓ(λ+n+k)Bp(λ−m+n+k, m−λ+µ)

Γ(λ−m+n+k)Γ(m−λ+µ) z^µ+n+k−1

= Γ(λ) Γ(µ)z^µ−1

∞

X

n,k=0

(λ)_n+k(α)n(β)_k (λ−m)_n+k

Bp(λ−m+n+k, m−λ+µ) B(λ−m, m−λ+µ)

(az)ⁿ n!

(bz)^k k!

= Γ(λ)

Γ(µ)z^µ−1F₁(λ, α, β;µ;az;bz;p).

Theorem 3.7. Let m−1< Re(λ−µ)< m < Re(λ), then D^λ−µ,p_z

z^λ−1(1−az)^−α(1−bz)^−β(1−cz)^−γ

= Γ(λ) Γ(µ)z^µ−1

∞

X

n,k,r=0

(λ)_n+k+r(α)n(β)_k(γ)r

(λ−m)_n+k+r

Bp(λ−m+n+k+r, µ−λ+m) B(λ−m, µ−λ+m)

(az)ⁿ n!

(bz)^k k!

(cz)^r r!

= Γ(λ)

Γ(µ)z^µ−1F_D,p³ (λ, α, β, γ;µ;az;bz;cz;p)

(3.4)

for |az |<1,|bz|<1 and |cz|<1.

Proof. Using the power series expansion of (1−az)^−α, (1−bz)^−β, (1−cz)^−γ and (2.6), we get D^λ−µ,p_z

z^λ−1(1−az)^−α(1−bz)^−β(1−cz)^−γ

=D_z^λ−µ,p ( _∞

X

n=0

∞

X

k=0

∞

X

r=0

(α)_n n!

(β)_k k!

(γ)_r

r! aⁿb^kc^rz^λ+n+k+r−1 )

=

∞

X

n,k,r=0

(α)_n n!

(β)_k k!

(γ)_r

r! aⁿb^kc^rD^λ−µ,p_z n

z^λ+n+k+r−1o

=

∞

X

n,k,r=0

(α)_n n!

(β)_k k!

(γ)_r

r! aⁿb^kc^rΓ(λ+n+k+r)B_p(λ−m+n+k+r, m−λ+µ)

Γ(λ−m+n+k+r)Γ(m−λ+µ) z^µ+n+k+r−1

= Γ(λ) Γ(µ)z^µ−1

∞

X

n,k,r=0

(λ)n+k+r(α)n(β)k(γ)r

(λ−m)_n+k+r

Bp(λ−m+n+k+r, m−λ+µ) B(λ−m, m−λ+µ)

(az)ⁿ n!

(bz)^k k!

(cz)^r r!

= Γ(λ)

Γ(µ)z^µ−1F_D,p³ λ, α, β, γ;µ;az;bz;cz;p).

Theorem 3.8. Let m−1< Re(λ−µ)< m < Re(λ) andm < Re(β)< Re(γ), then D^λ−µ,p_z

z^λ−1(1−z)^−α2F1(α, β;γ; x 1−z)

= Γ(λ) Γ(µ)z^µ−1

∞

X

n=0

∞

X

k=0

"

(α)n+k(β)n(λ)k

(β−m)_n(λ−m)_k Bp(β−m+n, γ−β+m)

B(β−m, γ−β+m)

Bp(λ−m+k, µ−λ+m) B(λ−m, µ−λ+m)

xⁿz^k n!k!

#

= Γ(λ)

Γ(µ)z^µ−1F2(α, β, λ;γ, µ;x, z;p)

(3.5)

for |x|+|z|<1.

Proof. Using the power series expansion of (1−z)^−α,F_p and (2.5), we get D^λ−µ,p_z

z^λ−1(1−z)^−α2F1(α, β;γ; x 1−z)

=D^λ−µ,p_z (

z^λ−1(1−z)^−α

∞

X

n=0

(α)n(β)n

(β−m)_nn!

Bp(β−m+n, γ−β+m) B(β−m, γ−β+m)

x 1−z

n)

=D^λ−µ,p_z (

z^λ−1(1−z)^−α−n

∞

X

n=0

(α)_n(β)_n (β−m)n

B_p(β−m+n, γ−β+m) B(β−m, γ−β+m)

xⁿ n!

)

=

∞

X

n=0

(α)_n(β)_n (β−m)_n

Bp(β−m+n, γ−β+m) B(β−m, γ−β+m)

xⁿ

n!D_z^λ−µ,pn

z^λ−1(1−z)^−α−no

= Γ(λ) Γ(µ)z^µ−1

∞

X

n=0

∞

X

k=0

"

(α)n+k(β)n(λ)k

(β−m)n(λ−m)_k

Bp(β−m+n, γ−β+m) B(β−m, γ−β+m)

Bp(λ−m+k, µ−λ+m) B(λ−m, µ−λ+m)

xⁿz^k n!k!

#

= Γ(λ)

Γ(µ)z^µ−1F2(α, β, λ;γ, µ;x, z;p).

4. Generating functions

In this section, we use the equalities (3.2), (3.3) and (3.5) for obtaining linear and bilinear generating relations for the extended hypergeometric function2F1.

Theorem 4.1. Let m−1< Re(λ−µ)< m < Re(λ), then

∞

X

n=0

(α)n

n! ²F1(α+n, λ;µ;z;p)tⁿ= (1−t)^−α2F1

α, λ;µ; z 1−t;p

, (4.1)

where |z|< min{1,|1−t|}.

Proof. Taking the identity

[(1−z)−t]^−α= (1−t)^−α

1− z 1−t

−α

in [11] and expanding the left hand side, we get

∞

X

n=0

(α)n

n! (1−z)^−α t

1−z n

= (1−t)^−α

1− z 1−t

−α

,

when |t| < |1−z|. If we multiply the both sides with z^λ−1 and apply the extended Caputo fractional derivative operatorDz^λ−µ,p, we get

D^λ−µ,p_z ( _∞

X

n=0

(α)ntⁿ

n! z^λ−1(1−z)^−α−n )

=D^λ−µ,p_z (

(1−t)^−αz^λ−1

1− z 1−t

−α) .

Since |t| < |1−z| and Re(λ) > Re(µ) > 0, it is possible to change the order of the summation and the derivative as

∞

X

n=0

(α)n

n! D_z^λ−µ,pn

z^λ−1(1−z)^−α−no

tⁿ= (1−t)^−αD_z^λ−µ,p (

z^λ−1

1− z 1−t

−α) .

So we get the result after using Theorem 3.5 on both sides.

Theorem 4.2. Let m−1< Re(λ−µ)< m < Re(λ), then

∞

X

n=0

(α)n

n! ²F₁(β−n, λ;µ;z;p)tⁿ= (1−t)^−αF₁

β, α, λ;µ;z; zt 1−t;p

,

where |t|< _1+|z|¹ .

Proof. Taking the identity

[1−(1−z)t]^−α= (1−t)^−α

1 + zt 1−t

−α

in [11] and expanding the left hand side, we get

∞

X

n=0

(α)_n

n! (1−z)ⁿtⁿ= (1−t)^−α

1− −zt 1−t

−α

,

when|t|<|1−z|. If we multiply the both sides withz^λ−1(1−z)^−β and apply the extended Caputo fractional derivative operatorDz^λ−µ,p, we get

D^λ−µ,p_z (_∞

X

n=0

(α)_n

n! z^λ−1(1−z)^−β(1−z)ⁿtⁿ )

=D^λ−µ,p_z (

(1−t)^−αz^λ−1(1−z)^−β

1− −zt 1−t

−α) .

Since |zt| < |1−t|and Re(λ) > Re(µ) > 0, it is possible to change the order of the summation and the derivative as

∞

X

n=0

(α)_n

n! D^λ−µ,p_z n

z^λ−1(1−z)^−β+no

tⁿ= (1−t)^−αD_z^λ−µ,p (

z^λ−1(1−z)^−β

1− −zt 1−t

−α) .

So we get the result after using Theorem 3.5 and Theorem 3.6.

Theorem 4.3. Let m−1< Re(β−γ)< m < Re(β) and m < Re(λ)< Re(µ), then

∞

X

n=0

(α)_n

n! ²F₁(α+n, λ;µ;z;p)₂F₁(−n, β;γ;u;p) =F₂

α, λ, β;µ, γ;z, ut 1−t;p

.

Proof. If we take t→(1−u)tin (4.1) and then multiply the both sides withu^β−1, we get

∞

X

n=0

(α)n

n! ²F₁(α+n, λ;µ;z;p)u^β−1(1−u)ⁿtⁿ=u^β−1[1−(1−u)t]^−α₂F₁

α, λ;µ; z

1−(1−u)t;p

.

Applying the fractional derivativeD^β−γu to both sides and changing the order we find

∞

X

n=0

(α)n

n! ²F1(α+n, λ;µ;z;p)D^β−γ_u n

u^β−1(1−u)ⁿ o

tⁿ

=D_u^β−γ

u^β−1[1−(1−u)t]^−α₂F₁

α, λ;µ; z

1−(1−u)t;p

when |z|<1,

1−u 1−zt

<1 and

z 1−t

+

ut 1−t

<1. If we write the equality like

∞

X

n=0

(α)_n

n! ²F₁(α+n, λ;µ;z;p)D^β−γ_u n

u^β−1(1−u)ⁿo tⁿ

=D_u^β−γ (

u^β−1

1− −ut 1−t

−α

2F₁ α, λ;µ; z 1−^−ut_1−t;p

!)

and using Theorem 3.5 and Theorem 3.8 we get the desired result.

5. Further results and observations

In this section, we apply the extended Caputo fractional derivative operator (3.1) to familiar functions e^z and ₂F₁(a, b;c;z). We also obtain the Mellin transforms of some extended Caputo fractional derivatives and we give the integral representations of extended hypergeometric functions.

Theorem 5.1. The extended Caputo fractional derivative of f(z) =e^z is D_z^µ,p{e^z}= z^m−µ

Γ(m−µ)

∞

X

n=0

zⁿ

n!Bp(m−µ, n+ 1) for allz.

Proof. Using the power series expansion of e^z and Theorem 3.3, we get D_z^µ,p{e^z}=

∞

X

n=0

1

n!D_z^µ,p{zⁿ}

=

∞

X

n=m

Γ(n+ 1)Bp(n−m+ 1, m−µ) Γ(n−µ+ 1)B(n−m+ 1, m−µ)

z^n−µ n!

=

∞

X

n=0

Γ(n+m+ 1)Bp(n+ 1, m−µ) Γ(n+m−µ+ 1)B(n+ 1, m−µ)

z^n+m−µ (n+m)!

= z^m−µ Γ(m−µ)

∞

X

n=0

zⁿ

n!B_p(m−µ, n+ 1).

Theorem 5.2. The extended Caputo fractional derivative of 2F1(a, b;c;z) is D_z^µ,p

2F1(a, b;c;z)

= (a)m(b)m

(c)_m

z^m−µ Γ(1−µ+m)

∞

X

n=0

(a+m)n(b+m)n

(c+m)_n(1−µ+m)_n

Bp(m−µ, n+ 1) zⁿ B(m−µ, n+ 1) for |z|<1.

Proof. Using the power series expansion of 2F1(a, b;c;z) and making similar calculations, we get D_z^µ,p{₂F₁(a, b;c;z)}=D^µ,p_z

( _∞ X

n=0

(a)_n(b)_n (c)_n

zⁿ n!

)

=

∞

X

n=0

(a)_n(b)_n

(c)nn! D_z^µ{zⁿ}

=

∞

X

n=m

(a)n(b)n

(c)_nn!

Γ(n+ 1)Bp(m−µ, n−m+ 1) Γ(n−µ+ 1)B(m−µ, n−m+ 1)z^n−µ

=

∞

X

n=0

(a)_n+m(b)_n+m (c)n+m(n+m)!

Γ(n+m+ 1)B_p(m−µ, n+ 1)

Γ(n+m−µ+ 1)B(m−µ, n+ 1)z^n+m−µ

= (a)m(b)m

(c)m

z^m−µ Γ(1−µ+m)

∞

X

n=0

(a+m)n(b+m)n

(c+m)n(1−µ+m)n

Bp(m−µ, n+ 1)zⁿ B(m−µ, n+ 1) . The following two theorems are about the Mellin transforms of extended Caputo fractional derivatives of two functions.

Theorem 5.3. Let Re(λ)> m−1 and Re(s)>0, then Mh

D_z^µ,pn z^λo

:si

= Γ(λ+ 1)Γ(s)

Γ(λ−m+ 1)Γ(m−µ)B(m−µ+s, λ−m+s+ 1)z^λ−µ. Proof. Using the definition of Mellin transform we get

M

D^µ,p_z n

z^λ o

:s

= Z ∞

0

p^s−1D_z^µ,p n

z^λ o

dp

= Z ∞

0

p^s−1 Γ(λ+ 1)Bp(m−µ, λ−m+ 1)

Γ(λ−µ+ 1)B(m−µ, λ−m+ 1)z^λ−µdp

= Γ(λ+ 1)z^λ−µ

Γ(λ−µ+ 1)B(m−µ, λ−m+ 1) Z ∞

0

p^s−1B_p(m−µ, λ−m+ 1)dp.

From the equality Z ∞

0

b^s−1B_p(x, y)db= Γ(s)B(x+s, y+s), Re(s)>0, Re(x+s)>0, Re(y+s)>0 in [2, page 21] we get the result.

Theorem 5.4. Let Re(s)>0 and |z|<1, then M

D^µ,p_z

(1−z)^−α :s

=Γ(s) z^m−µ Γ(m−µ)

∞

X

n=0

B(m−µ+s, n+s+ 1)

Γ(n+ 1) (µ)_nzⁿ.

Proof. With using the power series expansion of (1−z)^−α and takingλ=nin Theorem 5.3, we get M

D_z^µ,p

(1−z)^−α :s

=M

"

D^µ,p_z ( _∞

X

n=0

(α)_n n! zⁿ

) :s

#

=

∞

X

n=0

(α)_n

n! M[D^µ,p_z {zⁿ}:s]

= Γ(s) z^−µ Γ(m−µ)

∞

X

n=m

B(m−µ+s, n−m+s+ 1) Γ(n−m+ 1) (α)nzⁿ

= Γ(s) z^m−µ Γ(m−µ)

∞

X

n=0

B(m−µ+s, n+s+ 1)(α)_n+mzⁿ n! . Theorem 5.5. The following integral representations are valid

2F₁(a, b;c;z;p) = 1

B(b−m, c−b+m) Z 1

0

t^b−m−1(1−t)^c−b+m−1e

_−p

t(1−t)

2 F₁(a, b;b−m;zt)

dt, (5.1)

F1(a, b, c;d;x, y;p)

= 1

B(a−m, d−a+m) Z ₁

0

t^a−m−1(1−t)^d−a+m−1e

−p

t(1−t)

F₁(a, b, c;a−m;xt, yt)

dt, (5.2)

F₂(a, b, c;d, e;x, y;p)

= 1

B(b−m, d−b+m) Z 1

0

t^b−m−1(1−t)^d−b+m−1e

_−p

t(1−t)

F2(a, b, c;b−m, e;xt, y)

dt, (5.3)

F₂(a, b, c;d, e;x, y;p)

= 1

B(c−m, e−c+m) Z 1

0

t^c−m−1(1−t)^e−c+m−1e

_−p

t(1−t)

F2(a, b, c;d, b−m;x, yt)

dt, (5.4)

F2(a, b, c;d, e;x, y;p)

= 1

B(b−m, e−b+m)B(c−m, e−c+m) Z ₁

0

Z ₁

0

t^b−m−1u^c−m−1(1−t)^d−b+m−1(1−u)^e−c+m−1 e

−_t(1−t)^p −_u(1−u)^p

F₂(a, b, c;b−m, c−m;xt, yτ)

dtdu. (5.5) Proof. The integral representations (5.1)-(5.5) can be obtained directly by replacing the function B_p with its integral representation in (2.1)-(2.5), respectively.

Acknowledgements

This work was partly presented on September 3-7, 2012 at 1^stInternational Eurasian Conference on Math- ematical Sciences and Applications IECMSA-2012 at Prishtine-KOSOVO. This work was supported by Ahi Evran University Scientific Research Projects Coordination Unit. Project Number: PYO-FEN.4001.14.014.

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