A new representation of extended Mittage-Lefﬂer function and its properties

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A new representation of extended Mittage-Leffler

function and its properties

Abdul Ghaffar1*, Ayesha Saif1and Faheem Khan2

1 _{Department of Mathematical Science, BUITEMS Quetta, Pakistan; [email protected](A.}

Ghaffar)

1 _{Department of Mathematical Science, BUITEMS Quetta, Pakistan; [email protected]} 2 _{Department of Mathematics, University of Sargodha, Pakistan}

* Correspondence: [email protected]; Tel.: +92-3347138562 ‡ These authors contributed equally to this work.

Academic Editor: name

Version April 16, 2019 submitted to Preprints

Abstract:In this article, our main purpose is to establish a new extension of Mittag-Leffler function

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by using the known extended beta functionBω(a,b;p)introduced in [1]. It led to a novel extension of

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the applicability of Mittag-Leffler function that introduced them as distributions defined for a specific

3

set of functions. We also, investigate some of its important properties, namely recursion relation,

4

Mellin transform and differential formulas.

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Keywords:Mittag-Leffler function; extended Mittag-Leffler function; Extended beta function; Fox

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Wright function

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MSC:33B20, 33C20, 33C45, 33C60, 33B15, 33C05

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0. Introduction 9

Special functions are particular mathematical functions which are vital tools in higher calculus,

10

and generally play a key role in almost all branches of applied and pure mathematics. Many

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well-known special functions are helpful to solve boundary value problems. In addition, solutions

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of many differential equations come out in terms of special functions. The zoo of special function

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contains Gamma function, Beta function, Hypergeometric function, Bessel’s function, Mittag-Leffler

14

function and many more. Special functions have been one of the most popular area of mathematics. In

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17thcentury, beta function was investigated by Legendre and Euler. Beta function is commonly used

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in probability theory, physics, engineering and other areas of mathematics.

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Due to various implementations of generalized hypergeometric and Mittag-Leffler functions,

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many researchers have made their contribution to extend it in various forms. Recently, many authors

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[2–11] introduced several extensions of the well-known special functions due to their vital importance

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in mathematical and functional analysis.

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We start with the Gosta Mittag-Leffler functionEυ(z)[12] represented by the following series as:

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Eυ(z) =

∞

∑

k=0

zk

Γ(υk+1), (z,υ∈C;Re(υ)>0). (1) A more general form of Mittag-Leffler function given by Wiman [13] has the following form

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Eυ,τ(z) =

∞

∑

k=0

zk

Γ(υk+τ), (z,υ,τ∈C;Re(υ)>0,Re(τ)>0). (2)

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Throughout this article, we will denote byC,N,Z−0, andR+the set of Complex numbers, natural

24

numbers, non-positive integers and positive real numbers respectively. Prabhakar [14] established the

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generalization ofEυ,τ(z)in the following form

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Eλ υ,τ(z) =

∞

∑

k=0

(λ)_kzk

Γ(υk+τ)k!, (z,υ,τ,λ∈C;Re(υ)>0,Re(τ)>0), (3)

where(λ)krepresents the Pochhammer symbol [15] defined as follows

27

(λ)_k =

(

1; k=0,λ∈C− {0},

λ(λ+1)...(λ+k−1); k∈N,λ∈C = Γ(λ+k)

Γ(λ) λ∈C−Z

− 0.

Generalized Mittag-Leffler functionEλ,s

υ,τ(z)introduced by Prajapati and Shukla [16] has the following

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form

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Eλ,s υ,τ(z) =

∞

∑

k=0

(λ)skzk

Γ(υk+τ)k!, (z,υ,τ,λ∈C;Re(υ)>0,Re(τ)>0,Re(λ)>0;s∈(0, 1)∪N), (4)

where(λ)sk = Γ(_Γλ₍+_λsk₎) represents the generalized Pochhammer symbol. Ozarslan and Yilmaz [17]

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extended the Mittag-Leffler functionEλ,s

υ,τ(z)in the following way

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Eλ,s

υ,τ(z;p) =

∞

∑

k=0

Bp(λ+k,s−λ) B(λ,s−λ)

(s)k

Γ(υk+τ) zk

k!, (5)

where(z,υ,τ,λ∈C;Re(υ)>0,Re(τ)>0,Re(λ)>0;Re(p)>0,p=0;Re(s)>Re(λ)>0)and 32

Bp(a,b) = Z 1

0 u

a−1₍₁₋_u₎b−1_exp₋ p

u(1−u)

du, (Re(p)>0,Re(a)>0,Re(b)>0). (6)

is the extended Euler’s beta function [18] andB0(a,b) = B(a,b)whereB(a,b)is the classical beta

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function [15]. An interesting extension of extended beta functionBp(a,b)introduced by Parmar et al.

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[1] is of the following form

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Bω(a,b;p) = r

2p π

Z 1

0 u

a−3

2₍₁−_u₎b−32_K

ω+1₂

p u(1−u)

du, (ω∈C,Re(p)>0,Re(a)>0,Re(b)>0),

(7)

whereK_ω₊1

2 is the modified Bessel’s function. Motivated essentially by the demonstrated potential 36

for applications of these generalized Mittag-Leffler functions in many diverse areas of mathematical,

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physical, engineering and statistical sciences, we introduce here another interesting extension of the

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extended Mittag-Leffler function as follows

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Eλ,s;ω

υ,τ (z;p) =

∞

∑

k=0

Bω(λ+k,s−λ;p)

B(λ,s−λ)

(s)_k

Γ(υk+τ) zk

k!, (8)

where(z,υ,τ,λ,ω ∈C;Re(υ)>0,Re(τ) >0,Re(λ)>0;Re(p)>0;Re(s)> Re(λ) >0). For our 40

ambition, we remember the Fox- Wright functionrΨs[19]

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rΨs(z) =

∞

∑

k=0

Γ(α1+τ1k)...Γ(αr+τrk)

Γ(β1+µ1k)...Γ(βs+µsk)

zk

(3)

whereτiandµm∈R+such that 42

1+ s

∑

m=1

µm− r

∑

i=1

τi >0.

Further, making use of the extended Mittag-Leffler function (8). For each of these new extensions we

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obtain various integral representations, properties and Mellin transforms, together with differentiation,

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transformation, summation, generating function and asymptotic formulas.

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1. Integral Formulae 46

In this section, we obtain several integral representations of Mittag-Leffler function (Eλ,s;ω υ,τ (z;p))

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and consider certain special cases.

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Theorem 1. Let s,υ,τ,λ,∈CwithRe(s)>Re(λ)>0,Re(υ)>0,Re(τ)>0,and letRe(p)>0.Then 49

Eλ,s;ω

υ,τ (z;p) =

1

B(λ,s−λ)

r 2p

π ×

Z 1

0

uλ−3₂₍₁₋_u₎s−λ−3₂_K ω+1₂

_p

u(1−u)

Esυ,τ(uz)

du.

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Proof. By using equation (7) in (8), we get

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Eλ,s;ω

υ,τ (z;p) =

∞

∑

k=0

"r 2p

π Z 1

0 u

λ+k−3₂₍₁₋_u₎s−λ−3₂_K ω+1₂

p u(1−u)

du (s)_k

B(λ,s−λ)Γ(υk+τ) zk k! #

.

After interchanging the order of integration and summation in the above equation which is justified

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under the supposition of theorem (1) ,we obtain

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Eλ,s;ω

υ,τ (z;p) =

1

B(λ,s−λ)

r 2p

π Z 1

0

uλ−3₂₍₁₋_u₎s−λ−3₂_K ω+1₂

p u(1−u)

∞

∑

k=0

(s)_k

Γ(υk+τ) (zu)k

k! !

du.

With the help of equation (3) in above equation, we attain the desired consequence.

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Now, we obtain two different integral formulae of the extended MLF (10) as corollaries.

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Corollary 1. Let s,υ,τ,λ,∈CwithRe(s)>Re(˘)>0,Re(Æ)>0,Re(ø)>0and letRe(p)>0.Then 55

Eλ,s;ω

υ,τ (z;p) =

1

B(λ,s−λ)

r 2p

π Z ∞

0

yλ−3₂₍₁₊_y₎1−s_K ω+1₂

p(1+y)2 y

Es_υ,τ

yz

1+y

dy.

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Proof. If we setu= ₁₊y_y, in theorem (1),we acquire the consequence we wish to prove.

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Eλ,s;ω

υ,τ (z;p) =

2

B(λ,s−λ)

r 2p

π Z π₂

0

h

(sinψ)2(λ−1)(cosψ)2(s−λ−1)

K_ω₊1 2

p

sin2ψcos2ψ

Es_υ_,_τzsin2ψ

dy.

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Proof. If we setu=sin2ψ, in theorem (1), we acquire the consequence we want to prove. 58

2. Mellin Transform Formula 59

Theorem 2. Let s,υ,τ,λ,∈CwithRe(s)>Re(˘)>0,Re(Æ)>0,Re(ø)>0and letRe(p)>0.Then 60

the Mellin ransform formula for the extended MLF (??) is 61

M

Eλ,s;ω υ,τ (z;p);r

= 2

r−1

√

π

Γ(s+r−λ)

Γ(λ)Γ(s−λ)Γ

_r₋

ω

2

Γr+ω+1

2

×

2Ψ2



 

(s, 1),(λ+r, 1);

|z (τ,υ),(2r+s, 1);





. (13)

Proof. By definition, Mellin transform of the MLF is

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M

=

Z ∞

0 p

r−1_Eλ,s;ω

υ,τ (z;p)dp. From equation (10), we have

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M

=

Z ∞

0

"

pr−1 1 B(λ,s−λ)

r 2p

π Z 1

0 u

λ−3₂₍₁₋_u₎s−λ−3₂

K_ω₊1 2

_p

u(1−u)

Esυ,τ(uz)du

dp. (14)

By interchanging the order of integration in equation (14) that is verified by the assumptions of this

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theorem, we get

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M

= 1

B(λ,s−λ)

r 2

π Z 1

0

h

uλ−3₂₍₁₋_u₎s−λ−3₂_Es υ,τ(uz)

i

Z ∞

0 p

r−1 2_K

ω+1₂

p u(1−u)

dpdu. (15)

By puttingy= _u₍₁p₋_u₎in the internal integral in equation (15), we have

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M

= 1

B(λ,s−λ)

r 2

π Z 1

0 u

λ+r−1₍₁₋_u₎s+r−λ−1_Es υ,τ(uz) Z ∞

0 y

r−1 2_K

ω+1₂(y)dy

du. (16)

By using the consequence (Olveret al.(2010))

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Z ∞

0 y

r−1 2_K

ω+1₂(y)dy=2 r−3

2Γ₍r−ω

2 )Γ(

r+ω+1

(5)

and then applying definition (9) in equation (16), we have

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M

= 1

B(λ,s−λ)

2r−1 √

πΓ

r−ω

2

Γ

r+ω+1

2

Z 1

0 u

λ+r−1₍₁₋_u₎s+r−λ−1

∞

∑

k=0

(s)_k

Γ(υk+τ) (uz)k

k! du.

(17)

By interchanging the order of summation and integration and then applying the definition (2) in

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equation (17), we obtain

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M

= 2

r−1

√

π

Γ(s+r−λ)

Γ(λ)Γ(s−λ)Γ

r−ω

2

Γr+ω+1

2

∞

∑

k=0

Γ(s+k)Γ(λ+r+k)

Γ(τ+υk)Γ(2r+s+k) zk

k!. (18)

By the implementation of equation (9) in equation (18), we obtain the required consequence.

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the following result holds true 73

Z ∞

0 E

λ,s;ω

υ,τ (z;p)dp = 1 √

π

Γ(s+1−λ)

Γ(λ)Γ(s−λ)Γ

1−ω

2

Γ

ω+2

2

2Ψ2



 

(s, 1),(λ+1, 1);

|z (τ,υ),(2+s, 1);





. (19)

Proof. If we taker=1 in equation (13), we obtain the required result.

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3. Derivative Formulae 75

Theorem 3. Let s,υ,τ,λ,∈CwithRe(s)>Re(˘)>0,Re(Æ)>0,Re(ø)>0and letRe(p)>0.Then 76

the derivative formula for the extended MLF (8)is 77

dk dzkE

λ,s;ω

υ,τ (z;p) = (λ)kE

λ+k,s+k;ω

υ,kυ+τ (z;p), (k∈N∪ {0}). (20)

Proof. From equation (8), we have

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d dzE

λ,s;ω

υ,τ (z;p) =

∞

∑

k=1

Bω(λ+k,s−λ;p)

B(λ,s−λ)

(s)_k

Γ(υk+τ) zk−1 (k−1)!

!

= s ∞

∑

k=0

Bω(λ+1+k,s−λ;p)

B(λ,s−λ)

(s+1)_k

Γ(υk+υ+τ) zk k!

!

.

Using (5), we obtain

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d dzE

λ,s;ω

υ,τ (z;p) = λ

∞

∑

k=0

Bω(λ+1+k,s−λ;p)

B(λ+1,s−λ)

(s+1)k

Γ(υk+υ+τ) zk k!

!

.

In terms of (8), we get

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d dzE

λ,s;ω

υ,τ (z;p) = λE

(6)

Similarly,we have

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d2 dz2E

λ,s;ω

υ,τ (z;p) = (λ)2E

λ+2,s+2;ω υ,2υ+τ (z;p). By takingnthorder derivative of the MLF (9), we obtain the desired result.

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Theorem 4. Let s,υ,τ,λ,andη∈CwithRe(s)>Re(λ)>0,Re(υ)>0,Re(τ)>0and letRe(p)>0. 83

Then 84

dk

dzk

zτ−1_Eλ,s;ω υ,τ (ηz

υ_;_p₎₌_zτ−k−1_Eλ,s;ω υ,τ (ηz

υ_;_p₎_, ₍_k_∈

N∪ {0}). (21)

Proof. From equation (8), we get

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d dz

h

zτ−1_Eλ,s;ω

υ,τ (ηzν;p) i

= d

dz ∞

∑

k=0

Bω(λ+k,s−λ;p)

B(λ,s−λ)

(s)_k

Γ(υk+τ)(υk+τ−1)

ηkzυk+τ−1 k!

#

= zτ−2

∑

∞

k=0

Bω(λ+k,s−λ;p)

B(λ,s−λ)

(s)_k

Γ(υk+τ−1) (ηzυ₎k

k! . By using equation (8),above equation can be written as

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d dz

h

ν_;_p₎i₌_zτ−2_Eλ,s;ω υ,τ−1(ηz

ν_;_p₎_. Similarly, we have

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d2 dz2

h

ν_;_p₎i₌_zτ−3_Eλ,s;ω

υ,τ−2(ηzν;p).

After differentiating equation (8)ktimes, we get the desired result.

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4. Recurrence Relation 89

Theorem 5. Let s,υ,τ,λ,∈CwithRe(s)>Re(λ)>0,Re(υ)>0,Re(τ)>0and letRe(p)>0,then 90

Eλ,s;ω

υ,τ (z;p) =τE λ,s;ω

υ,τ+1(z;p) +υszE

λ+1,s+1;ω

υ,υ+τ+1 (z;p). (22)

Proof. By using (8) in the integrand of (10), we get

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Eλ,s;ω

υ,τ (z;p) =

1

B(λ,s−λ)

r 2p

π Z 1

0

uλ−3₂₍₁₋_u₎s−λ−3₂_K ω+1₂

p u(1−u)

τEs_υ_,_τ₊₁(uz)

du+ 1

B(λ,s−λ)

r 2p

π Z 1

0

h

uλ−3₂₍₁₋_u₎s−λ−3₂

K_ω₊1 2

_p

u(1−u)

υzd dzE

s

υ,τ+1(uz)

du.

(7)

Using equation (21) fork=1 in equation (23), we get

92

Eλ,s;ω

υ,τ (z;p) = τE λ,s;ω

υ,τ+1(z;p) +υsz 1

B(λ,s−λ)

r 2p

π

Z 1

0

h

u(λ+1)−3₂₍₁₋_u₎(s+1)−(λ+1)−3₂

K_ω₊1 2

_p

u(1−u)

Es_υ+_,_υ1₊_τ₊₁(uz)

du.

Using equation (10) in above equation, we obtain

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Eλ,s;ω

υ,τ (z;p) = τE λ,s;ω

υ,τ+1(z;p) +υszE

λ+1,s+1;ω υ,υ+τ+1 (z;p).

94

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