A new extension of the Riemann-Liouville fractional derivative operator

A NEW EXTENSION OF THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE OPERATOR GAUHAR RAHMAN, KOTTAKKARAN SOOPPY NISAR*, ZIVORAD TOMOVSKI

arXiv:1801.05001v1 [math.CA] 9 Jan 2018

Abstract. The main aim of this present paper is to present a new extension of the fractional derivative operator by using the extension of beta function recently defined by Shadab et al. Moreover, we establish some results related to the newly defined modified fractional derivative operator such as Mellin transform and relations to extended hypergeometric and Appell’s function via generating functions.

1. Introduction and preliminaries We begin with the well-known Riemann-Liouville (R-L) fractional derivative of order µ is defined (see [19],[7]) by Z x 1 µ Dx {f (x)} = f (t)(x − t)−µ−1 dt, ℜ(µ) > 0. (1.1) Γ(−µ) 0 For the case m − 1 < ℜ(µ) < m where m = 1, 2, · · · , it follows o dm µ−m n µ f (x) Dx {f (x)} = m Dx dx Z x o 1 dm n = m f (t)(x − t)−µ+m−1 dt , ℜ(µ) > 0 dx Γ(−µ + m) 0

(1.2)

and Dµx {xσ } =

Γ(σ + 1) xσ−µ , ℜ(σ) > −1. Γ(σ − µ + 1)

(1.3)

The researchers (see [6, 8, 13, 15, 21]) investigated the various extensions and generalization of fractional derivative operators. The extended R-L fractional derivative of order µ is defined in [11] by Z x 1 px2 µ −µ−1 Dx {f (x); p} = dt, ℜ(µ) > 0. (1.4) f (t)(x − t) exp − Γ(−µ) 0 t(x − t) For the case m − 1 < ℜ(µ) < m where m = 1, 2, · · · , it follows n o dm f (x); p Dµx {f (x); p} = m Dµ−m x dx Z x n dm px2 o 1 −µ+m−1 = m dt , ℜ(µ) > 0. f (t)(x − t) exp − dx Γ(−µ + m) 0 t(x − t) (1.5) 2010 Mathematics Subject Classification. 33C15, 33C05. Key words and phrases. Hypergeometric function, beta function, Extended hypergeometric function, Mellin transform, fractional derivative, Appell’s function. *Corresponding author.

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G. RAHMAN, K.S. NISAR, Z. TOMOVSKI

An extension of fractional derivative operator established in [1] is given by Z x px 1 qx µ Dx {f (x); p, q} = dt, ℜ(µ) > 0. (1.6) f (t)(x − t)−µ−1 . exp − − Γ(−µ) 0 t (x − t) For example, Dxµ {xν ; p, q}x=1 =

Bp,q (ν + 1, −µ) , Γ(−µ)

where Bp,q (x, y) is the extended beta function (see [10]) defined by Z 1 Bp,q (x, y) = tx−1 (1 − t)y−1 Ep,q (t)dt, x, y, p, q ∈ C, ℜ(p), ℜ(q) > 0, 0

q − pt − 1−t

where Ep,q (t) = e . For p = q we denote Bp,q by Bp and for p = q = 0, we get classical beta function defined by B(x, y) =

Z1

tx−1 (1 − t)y−1 dt, (ℜ(x) > 0, ℜ(y) > 0).

(1.7)

0

For the case m − 1 < ℜ(µ) < m where m = 1, 2, · · · , it follows o dm µ−m n µ f (x); p, q Dx {f (x); p, q} = m Dx dx Z x 1 dm n f (t)(x − t)−µ+m−1 = m dx Γ(−µ + m) 0 px qx o × exp − dt , ℜ(µ) > 0. − t (x − t)

(1.8)

Recently, Rahman et al. [17] define an extension of extended R-L fractional derivative of order µ as Z x 1 µ f (t)(x − t)−µ−1 Dx {f (x); p, q, λ, ρ} = Γ(−µ) 0 h h px i qx i ×1 F1 λ; ρ; − F λ; ρ; − dt, ℜ(µ) > 0. (1.9) 1 1 t (x − t) For the case m − 1 < ℜ(µ) < m where m = 1, 2, · · · , it follows n o dm Dµx {f (x); p, q, λ, ρ} = m Dµ−m f (x); p, q, λ, ρ x Zdxx h h m n qx i o px i d 1 F λ; ρ; − dt , f (t)(x − t)−µ+m−1 1 F1 λ; ρ; − = m 1 1 dx Γ(−µ + m) 0 t (x − t) (1.10) where ℜ(µ) > 0, ℜ(p) > 0 and ℜ(q) > 0. It is clear that when λ = ρ, then (1.9) reduce to (1.6). The Gauss hypergeometric function which is defined (see [18]) as 2 F1 (σ1 , σ2 ; σ3 ; z)

=

∞ X (σ1 )n (σ2 )n z n n=0

(σ3 )n

n!

, (|z| < 1),

(1.11)

A NEW EXTENSION..

3

σ1 , σ2 , σ3 ∈ C and σ3 6= 0, −1, −2, −3, · · · . The integral representation of hypergeometric hypergeometric function is defined by Z 1 Γ(σ3 ) tσ2 −1 (1 − t)σ3 −σ2 −1 (1 − zt)−σ1 dt, (1.12) 2 F1 (σ1 , σ2 ; σ3 ; z) = Γ(σ2 )Γ(σ3 − σ2 ) 0 ℜ(σ3 ) > ℜ(σ2 ) > 0, | arg(1 − z)| < π . The Apell series or bivariate hypergeometric series and its integral representation is respectively defined by ∞ X (σ1 )m+n (σ2 )m (σ3 )n xm y n ; F1 (σ1 , σ2 , σ3 ; σ4 ; x, y) = (σ4 )m+n m!n! m,n=0

for all σ1 , σ2 , σ3 , σ4 ∈ C, σ4 6= 0, −1, −2, −3, · · · ,

(1.13)

|x| < 1, |y| < 1.

Γ(σ4 ) Γ(σ1 )Γ(σ4 − σ1 ) Z1 × tσ1 −1 (1 − t)σ4 −σ1 −1 (1 − xt)−σ2 (1 − yt)−σ3 dt

F1 σ1 , σ2 , σ3 , σ4 ; x, y =

(1.14)

0

ℜ(σ4 ) > ℜ(σ1 ) > 0, | arg(1 − x)| < π and | arg(1 − y)| < π. Chaudhry et al. [2] introduced the extended beta function is defined by B(σ1 , σ2 ; p) = Bp (σ1 , σ2 ) =

Z1

p

tσ1 −1 (1 − t)σ2 −1 e− t(1−t) dt

(1.15)

0

(where ℜ(p) > 0, ℜ(σ1 ) > 0, ℜ(σ2 ) > 0) respectively. When p = 0, then B(σ1 , σ2 ; 0) = B(σ1 , σ2 ). The extended hypergeometric function introduced in [3] by using the definition of extended beta function Bp (δ1 , δ2 ) as follows: Fp (σ1 , σ2 ; σ3 ; z) =

∞ X Bp (σ2 + n, σ3 − σ2 ) n=0

B(σ2 , σ3 − σ2 )

(σ1 )n

zn , n!

(1.16)

where p ≥ 0 and ℜ(σ3 ) > ℜ(σ2 ) > 0, |z| < 1. In the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as 1 B(σ2 , σ3 − σ2 ) Z 1 × tσ2 −1 (1 − t)σ3 −σ2 −1 (1 − zt)−σ1 exp

Fp (σ1 , σ2 ; σ3 ; z) =

−p dt, t(1 − t) 0 p ≥ 0, ℜ(σ3 ) > ℜ(σ2 ) > 0, | arg(1 − z)| < π .

(1.17)

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The extended Appell’s function is defined by (see [11]) F1 (σ1 , σ2 , σ3 ; σ4 ; x, y; p) =

∞ X Bp (σ1 + m + n, σ4 − σ1 ) n=0

B(σ1 , σ4 − σ1 )

(σ2 )m (σ3 )n

xm y n m!n!

(1.18)

where p ≥ 0 and ℜ(σ4 ) > ℜ(σ1 ) > 0 and |x|, |y| < 1. They [11] defined its integral representation by 1 F1 (σ1 , σ2 , σ3 ; σ4 ; z; p) = B(σ1 , σ4 − σ1 ) Z 1 σ1 −1 σ4 −σ1 −1 −σ2 −σ3 × t (1 − t) (1 − xt) (1 − yt) exp

−p dt, t(1 − t) 0 (1.19) ℜ(p) > 0, ℜ(σ4 ) > ℜ(σ1 ) > 0, | arg(1 − x)| < π, | arg(1 − y)| < π .

It is clear that when p = 0, then the equations (1.16)-(1.19) reduce to the well known hypergeometric, confluent hypergeometric and Appell’s series and their integral representation respectively. Very recently Shadab et al. [5] introduced a new and modified extension of beta function as: Z 1 p α Bp (σ1 , σ2 ) = tσ1 −1 (1 − t)σ2 −1 Eα − dt, (1.20) t(1 − t) 0 where ℜ(σ1 ) > 0, ℜ(σ2 ) > 0 and Eα . is Mittag-Leffler function defined by ∞ X Eα z = n=0

zn . Γ(αn + 1)

(1.21)

Obviously, when α = 1 then Bp1 (x, y) = Bp (x, y) is the extended beta function (see[2]). Similarly, when when α = 1 and p = 0, then B01 (x, y) = B0 (x, y) is the classical beta function. They [5] also defined extended hypergeometric function and its integral representation ∞ X Bpα (σ2 + n, σ3 − σ2 ) z n α Fp (σ1 , σ2 ; σ3 ; z) = 2 F1 σ1 , σ2 ; σ3 ; z; p, α = (σ1 )n B(σ2 , σ3 − σ2 ) n! n=0 ∞ X

B(σ2 + n, σ3 − σ2 ; p, α) z n = (σ1 )n B(σ2 , σ3 − σ2 ) n! n=0 where p, α ≥ 0, σ1 , σ2 , σ3 ∈ C and |z| < 1. 1 Fpα (σ1 , σ2 ; σ3 ; z) = β(σ2 ; σ3 − σ2 ) Z 1 × tσ2 −1 (1 − t)σ3 −σ2 −1 (1 − tz)−σ1 Eα − 0

p dt, t(1 − t)

(1.22)

(1.23)

where ℜ(p) > 0, ℜ(α) > 0, ℜ(σ3 ) > ℜ(σ2 ) > 0. Obviously when α = 1, then the hypergeometric function (1.22) will reduce to the extended hypergeometric function (1.16) and similarly when α = 1 and p = 0 then the hypergeometric function (1.22) will reduce to

A NEW EXTENSION..

5

the hypergeometric function (1.11). For various extensions and generalizations of beta function and hypergeometric functions the interested readers may refer to the recent work of researchers (see e. g., [4, 9, 11, 12]). 2. extension of Appell’s functions and its integral representations In this section, we used the definition (1.20) and consider the following modified extension Appell’s functions. Definition 2.1. The modified extended Appell’s function F1 is defined by α F1,p (σ1 , σ2 , σ3 ; σ4 ; x, y) = F1 (σ1 , σ2 , σ3 ; σ4 ; x, y; p, α)

=

∞ X

(σ2 )m (σ3 )n

m,n=0

Bpα (σ1 + m + n, σ4 − σ1 ) xm y n B(σ1 , σ4 − σ1 ) m! n!

∞ X

B(σ1 + m + n, σ4 − σ1 ; p, α) xm y n = (σ2 )m (σ3 )n B(σ1 , σ4 − σ1 ) m! n! m,n=0

(2.1)

where p, α ≥ 0, σ1 , σ2 , σ3 , σ4 ∈ C and |x| < 1, |y| < 1. Remark 2.1. Setting α = 1 in (2.1), then we get the extended Appell’s functions (see [11]). Theorem 2.1. The following integral representation holds true for (2.1) Z 1 1 F1 σ1 , σ2 , σ3 ; σ4 ; x, y; p, α = tσ1 −1 (1 − t)σ4 −σ1 −1 (1 − tx)−σ2 (1 − ty)−σ3 B(σ1 ; σ4 − σ1 ) 0 p × Eα − dt, (2.2) t(1 − t) where ℜ(p) > 0, ℜ(α) > 0, ℜ(σ4 ) > ℜ(σ1 ) > 0. Proof. Using the definition (1.20) in (2.1), we have Z 1 1 tσ1 −1 (1 − t)σ4 −σ1 −1 B(σ1 ; σ4 − σ1 ) 0 ∞ X (σ2 )m (σ3 )n (tz)m (tz)n dt. × m!n! m,n=0

F1 σ1 , σ2 , σ3 ; σ4 ; x, y; p, α =

(2.3)

Since ∞ X (σ2 )m (σ3 )n (z)m (z)n = (1 − tz)−σ2 (1 − tz)−σ3 . m!n! m,n=0

Thus by using (2.4) in (2.3), we get the desired result.

(2.4)

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3. Extension of fractional derivative operator In this section, we define a new and modified extension of Riemann-Liouville fractional derivative and obtain its related results. Definition 3.1. Dµ;α z;p {f (z)}

1 = Γ(−µ)

Z

z

0

f (t)(z − t)−µ−1 Eα −

pz 2 dt, ℜ(µ) > 0. t(z − t)

(3.1)

For the case m − 1 < ℜ(µ) < m where m = 1, 2, · · · , it follows o dm µ−m;α n D f (z) Dµ;α {f (z)} = z;p dz m z;p Z z dm n 1 pz 2 o −µ+m−1 = m dt , ℜ(µ) > 0. (3.2) f (t)(z − t) Eα − dz Γ(−µ + m) 0 t(z − t) Remark 3.1. Obviously if α = 1, then (3.1) and (3.2) respectively reduces to the extended fractional derivative (1.4) and (1.5). Similarly, if we set α = 1 and p = 0 we get (1.1) and (1.2). Now, we prove some theorems involving the modified extension of fractional derivative operator. Theorem 3.1. The following formula hold true, Dµz {z η ; p, α}

Bpα (η + 1, −µ) η−µ = z , ℜ(µ) > 0. Γ(−µ)

(3.3)

Proof. From (3.1), we have Dµz {z η ; p, α}

1 = Γ(−µ)

Z

z η

t (z − t)

−µ−1

0

Eα

pz 2 dt. − t(z − t)

(3.4)

Substituting t = uz in (3.4), we get Z 1 1 pz 2 µ η η −µ−1 Dz {z ; p, α} = dt (uz) (z − uz) Eα − Γ(−µ) 0 uz(z − uz) Z 1 p z η−µ dt, uη (1 − u)−µ−1 Eα − = Γ(−µ) 0 u(1 − u) In view of (1.20) to the above equation, we get the required result.

Theorem 3.2. Let ℜ(µ) > 0 and assume that the function f (z) is analytic at the origin P n + with its Maclaurin expansion given by f (z) = ∞ n=0 an z where |z| < δ for some δ ∈ R . Then ∞ X µ;α µ an Dµz {z n ; p, α} Dz;p {f (z)} = Dz {f (z); p, α} = n=0

∞

1 X = an Bpα (n + 1, −µ)z n−µ . Γ(−µ) n=0

(3.5)

A NEW EXTENSION..

7

Proof. Using the series expansion of the function f (z) in (3.1) gives Z zX ∞ 1 pz 2 µ Dz {f (z); p, α} = dt. an tn (z − t)−µ−1 Eα − Γ(−µ) 0 n=0 t(z − t) The series is uniformly convergent on any closed disk centered at the origin with its radius smaller than δ, so does on the line segment from 0 to a fixed z for |z| < δ. Thus it guarantee terms by terms integration as follows ∞ n 1 Z z X pz 2 µ dt tn (z − t)−µ−1 Eα − Dz {f (z); p, α} = an Γ(−µ) uz(z − uz) 0 n=0 ∞ X

=

an Dµz {z n ; p, α}.

n=0

Now, applying Theorem 3.1, we get ∞

Dµz {f (z); p, α}

1 X = an Bpα (n + 1, −µ)z n−µ , ℜ(µ) > 0. Γ(−µ) n=0

which is the required proof.

Example 3.1. The following result holds true: ∞ zn z −µ X α µ z Bp (n + 1, −µ) . Dz {e ; p, α} = Γ(−µ) n=0 n!

(3.6)

Using the power series of exp(z) and applying Theorem 3.2, we have Dµz {ez ; p, α}

∞ X 1 µ n D {z ; p, α}. = n! z n=0

(3.7)

Now, applying Theorem 3.1, we get the desired result. Theorem 3.3. The following formula holds true: Dη−µ {z η−1 (1 − z)−β ; p, α} = z

Γ(η) µ−1 α z 2 F1;p β, µ; η; z , Γ(µ)

(3.8)

where ℜ(µ) > ℜ(η) > 0 and |z| < 1. Proof. By direct calculation, we have Dη−µ {z η−1 (1 z

− z)

−β

Z z 1 pz 2 ; p, α} = dt tη−1 (1 − t)−β (z − t)η−µ−1 Eα − Γ(η − µ) 0 t(z − t) Z z t η−µ−1 pz 2 z µ−η−1 η−1 −β dt. t (1 − t) (1 − ) Eα − = Γ(η − µ) 0 z t(z − t)

Substituting t = zu in the above equation, we get Z 1 pz 2 z µ−1 η−µ η−1 −β du uη−1 (1 − uz)−β (1 − u)η−µ−1 Eα − Dz {z (1 − z) ; p, α} = Γ(η − µ) 0 uz(z − uz) Z 1 z µ−1 p = du uη−1 (1 − uz)−β (1 − u)η−µ−1 Eα − Γ(η − µ) 0 u(1 − u) Using (1.23) and after simplification we get the required proof.

8


Theorem 3.4. The following formula holds true: Dη−µ {z η−1 (1 z

− az)

−α

(1 − bz)

−β

Γ(η) µ−1 ; p, α} = z F1 η, α, β; µ; az, bz; p, α , Γ(µ)

(3.9)

where ℜ(µ) > ℜ(η) > 0, ℜ(α) > 0, ℜ(β) > 0, |az| < 1 and |bz| < 1. Proof. Consider the following power series expansion ∞ X ∞ X (az)m (bz)n −α −β (1 − az) (1 − bz) = (α)m (β)n . m! n! m=0 n=0 Now, applying Theorem 3.3, we obtain Dη−µ {z η−1 (1 − az)−α (1 − bz)−β ; p, α} z =

∞ X ∞ X

m=0 n=0

(α)m (β)n

(a)m (b)n η−µ η+m+n−1 D {z ; p, α}. m! n! z

Using Theorem 3.1, we have Dη−µ {z η−1 (1 − az)−α (1 − bz)−β ; p, α} z ∞ X ∞ X (a)m (b)n Bpα (η + m + n, µ − η) µ+m+n−1 z . = (α)m (β)n m! n! Γ(µ − η) m=0 n=0

Now, applying (2.1), we get Dη−µ {z η−1 (1 − az)−α (1 − bz)−β ; p, α} = z

Γ(η) µ−1 z F1 η, α, β; µ; az, bz; p, α . Γ(µ)

Theorem 3.5. The following Mellin transform formula holds true: n o π z η−µ η M Dµ;α (z ); p → r, = B(η + r + 1, −µ + r), z;p sin(πr) Γ(−µ)Γ(1 − rα)

(3.10)

where ℜ(η) > −1, ℜ(µ) > 0, ℜ(r) > 0, ℜ(r) > 0. Proof. Applying the Mellin transform on definition (3.1), we have n o Z ∞ µ;α η η M Dz;p (z ); p → r = pr−1 Dµ;α z;p (z )dp 0 Z ∞ nZ z 1 pz 2 o r−1 = dt dp p tη (z − t)−µ−1 Eα − Γ(−µ) 0 t(z − t) 0 Z Z pz 2 o t −µ−1 z −µ−1 ∞ r−1 n z η dt dp p Eα − t (1 − ) = Γ(−µ) 0 z t(z − t) 0 Z ∞ o nZ 1 z η−µ p r−1 = du dp p uη (1 − u)−µ−1 Eα − Γ(−µ) 0 u(1 − u) 0 From the uniform convergence of the integral, the order of integration can be interchanged. Thus, we have n o z η−µ η M Dµ;α (z ); p → r = z;p Γ(−µ)

A NEW EXTENSION..

×

Z

1

uη (1 − u)−µ−1

0

Letting v = M

n

p , u(1−u)

η Dµ;α z;p (z ); p

Z

∞

0

9

pr−1 Eα −

p dp du. u(1 − u)

(3.11)

(3.11) reduces to z η−µ →r = Γ(−µ) o

Z

1

u

η+r

(1 − u)

−µ+r−1

0

By using the following formula, Z ∞ δ v r−1 Eα,γ (−wv)dv = 0

Z

∞ 0

v r−1 Eα − v dv du.

Γ(r)Γ(δ − r) , Γ(δ)w r Γ(γ − rα)

(3.12)

for γ = δ = 1 and w = 1, we have n o z η−µ Γ(r)Γ(1 − r) Z 1 µ;α η M Dz;p (z ); p → r = uη+r (1 − u)−µ+r−1 du Γ(−µ)Γ(1 − rα) 0 z η−µ Γ(r)Γ(1 − r) = B(η + r + 1, −µ + r), Γ(−µ)Γ(1 − rα) Now, using the Euler’s reflection formula on Gamma function, π Γ(r)Γ(1 − r) = , sin(πr)

(3.13)

we get the desired result.

Theorem 3.6. The following Mellin transform formula holds true: n o π B(1 + r, −µ + r) µ;α α M Dz;p ((1 − z) ); p → r = z −µ sin(πr) Γ(−µ)Γ(1 − rα) ×2 F1 λ, r + 1; 1 − µ + 2r; z ,

(3.14)

where ℜ(p) > 0, ℜ(µ) < 0, ℜ(r) > 0. Proof. Applying Theorem 3.5 with η = n, we can write M

n

Dµ;α z;p ((1

λ

o

− z) ); p → r =

∞ X (λ)n n=0

n!

M

n

n Dµ;α z;p (z ); p

→r

o

∞

Γ(r)Γ(1 − r) X (λ)n = B(n + r + 1, −µ + r)z n−µ Γ(−µ)Γ(1 − rα) n=0 n! ∞

=z

−µ

Γ(r)Γ(1 − r) X (λ)n z n . B(n + r + 1, −µ + r) Γ(−µ)Γ(1 − rα) n=0 n!

In view of (3.13), we obtain the required result.

4. Generating relations In this section, we derive generating relations of linear and bilinear type for the extended hypergeometric functions.

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Theorem 4.1. The following generating relation holds true: ∞ X (λ)n α z n −λ α , F λ + n, β; γ; z t = (1 − t) F λ, β; γ; 2 1;p 2 1;p n! 1 − t n=0

(4.1)

where |z| < min(|1, 1 − t|), ℜ(α) > 0, ℜ(γ) > ℜ(β) > 0. Proof. Consider the following series identity [(1 − z) − t]−λ = (1 − t)−λ [1 −

z −λ ] . 1−t

Thus, the power series expansion yields ∞ t n X (λ)n z −λ (1 − z)−λ ] . = (1 − t)−λ [1 − n! 1 − z 1 − t n=0

(4.2)

β−γ;α Multiplying both sides of (4.2) by z β−1 and then applying the operator Dz;p on both sides, we have ∞ t n i h hX z −λ i (λ)n −λ β−1 −λ β−γ;α β−1 β−γ;α . (1 − z) z = (1 − t) Dz;p z 1− Dz;p n! 1−z 1−t n=0 β−γ;α Interchanging the order of summation and the operator Dz;p , we have ∞ i h X (λ)n β−γ;α h β−1 z −λ i β−γ;α . Dz;p z (1 − z)−λ−n tn = (1 − t)−λ Dz;p z β−1 1 − n! 1 − t n=0

Thus by applying Theorem 3.3, we obtain the required result. Theorem 4.2. The following generating relation holds true: ∞ X zt (β)n α n −β F δ − n, β; γ; z t = (1 − t) F λ, δ, β; γ; − ; p, α , 2 1;p 1 n! 1 − t n=0 where |t|
0, ℜ(β) > 0, ℜ(γ) > ℜ(λ) > 0.

Proof. Consider the series identity h zt i−β [1 − (1 − z)t]−β = (1 − t)−β 1 + . 1−t Using the power series expansion to the left sides, we have ∞ h X −zt i−β (β)n (1 − z)n tn = (1 − t)−β 1 − . n! 1 − t n=0

(4.4)

Multiplying both sides of (4.4) by z α−1 (1 − z)−δ and applying the operator Dλ−γ;α on z;p both sides, we have ∞ h hX i −zt −β i (β)n α−1 λ−1 −δ −β λ−γ;α λ−γ;α −δ+n n 1− , z (1 − z) Dz;p z (1 − z) t = (1 − t) Dz;p n! 1−t n=0 where ℜ(λ) > 0 and |zt| < |1 − t|, thus by Theorem 2.1, we have ∞ h i X −zt −β i (β)n λ−γ;α h λ−1 λ−1 −δ 1 − z (1 − z) . z (1 − z)−δ+n tn = (1 − t)−β Dλ−γ;α Dz;p z;p n! 1−t n=0

A NEW EXTENSION..

11

Applying Theorem 3.4 on both sides, we get the required result. Theorem 4.3. The following result holds true: ∞ h i z µ−1 X (µ)n zn µ η−µ;α z η−1 Eγ,δ (z) = Dz;p Bpα (η + n, µ − η) , Γ(µ − η) n=0 Γ(γn + δ) n!

(4.5)

µ where γ, δ, µ, α ∈ C, ℜ(p) > 0, ℜ(µ) > ℜ(η) > 0, ℜ(λ) > 0, ℜ(ρ) > 0 and Eγ,δ (z) is Mittag-Leffler function (see [14]) defined as: µ Eγ,δ (z)

=

∞ X n=0

(µ)n z n . Γ(γn + δ) n!

(4.6)

Proof. Using (4.6) in (4.5), we have η−µ;α Dz;p

h

z

η−1

µ Eγ,δ (z)

i

=

η−µ;α Dz;p

∞ h nX η−1 z n=0

By Theorem 2.1, we have ∞ h i X η−µ;α η−1 µ Dz;p z Eγ,δ (z) = n=0

(µ)n z n oi . Γ(γn + δ) n!

(µ)n n η−µ;α h η+n−1 io Dz;p z . Γ(γn + δ)

Applying Theorem 3.1, we get the required proof.

Remark 4.1. In [10] Mehrez and Tomovski introduced a new p-Mittag-Leffler function defined by (µ,η,ω) Eλ,γ,δ;p (z)

=

∞ X k=0

Bp (η + k, ω − η) z k (µ)k , (z ∈ C) [Γ(γk + δ)]λ B(η, ω − η) k!

(4.7)

(µ, η, ω, γ, δ, λ > 0, ℜ(p) > 0). Hence we get the following corollary. µ Corollary 4.1. The following formula holds true for Eγ,δ (z): n o Γ(η) (µ,η,µ) η−µ η−1 µ Dz z Eγ,δ (z); p = z µ−1 E (z), Γ(µ) 1,γ,δ;p

(4.8)

(µ, η, γ, δ > 0, ℜ(p) > 0). Theorem 4.4. The following result holds true:   (α , A ) ; i i 1,m o n z µ−1   η−µ;α Dz;p z η−1 m Ψn  |z  = Γ(µ − η) (βj , Bj )1,n ; ∞ Qm X Γ(αi + Ai k) α zk Qni=1 × Bp (η + k, µ − η) , k! j=1 Γ(βj + Bj k) k=0

(4.9)

12


where ℜ(p) > 0, ℜ(α) > 0, ℜ(µ) > ℜ(η) > 0, ℜ(λ) > 0, ℜ(ρ) > 0 and m Ψn (z) represents the Fox-Wright function (see [7], pp. 56-58)   (αi , Ai )1,m ; ∞ Qm   X Q i=1 Γ(αi + Ai k) z k . (4.10) |z  = m Ψn (z) = m Ψn  n Γ(β j + Bj k) k! j=1 k=0 (βj , Bj )1,n ; Proof. Applying Theorem 3.1 and followed the same procedure used in Theorem 4.3, we get the desired result. Remark 4.2. In [20] Sharma and Devi introduced extended Wright generalized hypergeometric function defined by   (αi , Ai )1,m , (γ, 1);   |(z; p)  m+1 Ψn+1 (z; p) = m+1 Ψn+1  (βj , Bj )1,n ; (c, 1) Q ∞ m k X 1 i=1 Γ(αi + Ai k) Bp (γ + k, c − γ)z Q = , (4.11) n Γ(c − γ) k! j=1 Γ(βj + Bj k) k=0

(ℜ(p) > 0, ℜ(c) > ℜ(γ) > 0).

Hence we get the following corollary. Corollary 4.2. The following results holds true for the Fox-Wright hypergeometric function:     (α , A ) ; (α , A ) , (η, 1); i i 1,m i i 1,m n o     η−1 µ−1 Dη−µ z Ψ Ψ ; p = z |z |(z; p)  ,  m n m+1 n+1  z (βj , Bj )1,n ; (βj , Bj )1,n ; (µ, 1) (4.12) (ℜ(p) > 0, ℜ(µ) > ℜ(η) > 0). 5. Concluding remarks In this paper, we established a modified extension of Riemnn-Liouville fractional derivative operator. We conclude that when α = 1, then all the results established in this paper will reduce to the results associated with classical Riemnn-Liouville derivative operator (see [11]). Similarly, if we letting α = 1 and p = 0 then all the results established in this paper will reduce to the results associated with classical Riemnn-Liouville fractional derivative operator (see [7]). References [1] D. Baleanu, P. Agarwal, R. K. Parmar, M. M. Alquarashi, S. Salahshour, Extension of the fractional derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10(2017), 2914-2924.

A NEW EXTENSION..

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[2] M. A. Chaudhry, A. Qadir, M. Rafique, S. M. Zubair,Extension of Eulers beta function, J. Comput. Appl. Math. 78 (1997) 1932. [3] M. A. Chaudhry, A. Qadir, H. M. Srivastava and R. B. Paris, Extended Hypergeometric and Confluent Hypergeometric functions, Appl. Math. Comput., 159 (2004) 589-602 [4] J. Choi, A. K. Rathie, R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Mathematical J. 36 (2014), No. 2, pp. 357-385. [5] M. Shadab, S. Jabee and J. Choi, An extension of beta function and its application, Far East Journal of Mathematical Sciences, 103(2018), No. 1, pp 235-251. [6] I. O. Kiymaz, A. Cetinkaya, P. Agarwal, An extension of Caputo fractional derivative operator and iyts application, J. Nonlinear Sci. Appl., 9 (2016), 3611-3621. [7] A. A. Kilbas, H. M. Sarivastava, J. J. Trujillo, Theory and application of fractional differential equation, North-Holland Mathematics Studies, Elsevier Sciences B.V., Amsterdam, (2006). [8] M. J. Luo, G. V. Milovanovic, P. Agarwal, Some results on the extended beta and extended hypergeometric functions, Appl. Math. Comput., 248 (2014), 631-651. [9] S. Mubeen, G. Rahman, K. S. Nisar, J. Choi. M. Arshad, An extended beta function and its properties, Far East Journal of Mathematical Sciences, 102(2017), 1545-1557. [10] K.Mehrez, Z.Tomovski, On a new (p,q)-Mathieu type power series and its applications, submitted ¨ ¨ [11] M. A. Ozerslan, E. Ozergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Mathematical and Computer Modelling, 52 (2010), 1825-1833. ¨ ¨ [12] M. A. Ozerslan, E. Ozergin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), 4601-4610. [13] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark (eds.), NIST handbook of mathematical functions, With 1 CD-ROM (Windows, Macintosh and UNIX), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, (2010). [14] T.R. Prabhakar, A singular integral equation with a generalized MittagLeffler function in the kernel, Yokohama Math. J. 19, 715, (1971). [15] R. K. Parmar, Some generating relations for generalized extended hypergeometric functions involving generalized fractional derivative operator, J. Concr. Appl. Math., 12 (2014), 217-228. [16] P. I. Pucheta, A new extended beta function, International Journal of Mathematics And its Applications Volume 5, Issue 3-C (2017), 255-260. [17] G. Rahman, S. Mubeen, K. S. Nisar, J. Choi, Extended special functions and fractional integral operator via an extended Beta function, Submitted. [18] E. D. Rainville, Special functions, The Macmillan Company, New York, 1960. [19] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Niko´lskiˇ, Translated

14


from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon, (1993). [20] S. C. Sharma, M. Devi, Certain properties of extended Wright generalized hypergeometric function , Annals of Pure and Applied Mathematics Vol. 9, No. 1, 2015, 45-51. [21] H. M. Srivastava, R. K. Parmar, P. Chopra, A class of extended fractional derivative operators and associated generating relations involving hypergeometric functions, Axioms, 1 (2012), 238-258. Gauhar Rahman: Department of Mathematics, International Islamic University, Islamabad, Pakistan E-mail address: [email protected] Kottakkaran Sooppy Nisar: Department of Mathematics, College of Arts and ScienceWadi Aldawaser, 11991, Prince Sattam bin Abdulaziz University,Alkharj, Kingdom of Saudi Arabia E-mail address: [email protected]; [email protected] Zivorad Tomovski: University St. Cyril and Methodius, Faculty of Natural Sciences and Mathematics, Institute of Mathematics, Repubic of Macedonia. E-mail address: [email protected]