Vol 8, No 5 (2017)

Available online throug

ISSN 2229 – 5046

FRACTIONAL INTEGRAL TRANSFORMATIONS OF MITTAG- LEFFLER TYPE E-FUNCTION

WITH MULTIVARIABLE POLYNOMIAL AND ALEPH FUNCTION

Dr. SANJAY BHATTER*, RAKESH KUMAR BOHRA

*Department of Mathematics,

Malaviya National Institute of Technology, Jaipur, 302017, India.

Department of Mathematics,

Global College of Technology, Jaipur, 302022, India.

(Received On: 13-04-17; Revised & Accepted On: 17-05-17)

ABSTRACT

I

n Present paper, we study of various fractional integral transformations of Mittag-Leffer type E-function with Multivariable Polynomial

(

)

l l _,...,x

1 x U ,..., 1 U V

S and Aleph function in series. Also find results for earlier defined

Mittag- Leffler type functions.

Key words: Aleph functions in series, Multivariable Polynomial, Fractional Integral transformations, Mittag- Leffler

type function.

1. INTRODUCTION

It is found by recent works of several authors that the Mittag-Leffler (M-L) function is the solution of fractional differential and integral equations. Unified M-L function named by E-function [1]. Now we will study Erdelyi-Kober, Riemann Liouville and other fractional integral transformation of newly defined M-L type E-function.

In this Paper we use the following definitions

Riemann-Liouville Fractional integral Operator (I_cθ₊Ψ) ( )x [6]

dt

x

c

(

t

)

1

)

t

x

(

1

)

x

)(

c

I

(

∫

−

θ

−

Ψ

θ

Γ

=

Ψ

θ

+

(1) Where

θ

∈

C

;

ℜ

( )

θ

>

0.

Erdelyi-Kober fractional integral operator η,θ_)(x)

0

(Ξ ₊ [6]

( )

_{( ) (}

x

)

f

( )

t

dt

0

θ

t

1

η

t

x

η

Γ

θ

η

x

x

f

θ

η,

0

Ξ

∫

−

−

−

−

=













+

(2)

Where η,θ∈C;ℜ

( )

η >0andℜ

( )

θ >0.

The Multivariable polynomial

(

)

l l _,...,x

1 x U ,..., 1 U V

S introduced by Srivastava and Garg (1987) [8, p.686, eq. (1.4)] is defined in the following manner:

(

)

(

)

U R V

i i

U ,...,U₁ i 1

S_V (x , ..., x ) V A V, R , ..., R

1 _{R ,...,R 0} 1 _!

1 U R_{i i}

i 1

l

R_i xi l

l _l l _R

i l

≤ ∑

=

= ∑ −

=

∑ =

(3)

Where

_V

₌

_0,1,2

_…

and

l

U

,...,

1

U

an arbitrary positive integers and the coefficients

_,...,

_R

₎

1

R

,

V

(

A

_l

are arbitrary constants (real or complex).

Fractional Integral Transformations of Mittag- Leffler Type E-function with Multivariable Polynomial…. / IJMA- 8(5), May-2017.

In 1903, Gosta Mittag-Leffler [5], introduced the function

E

_α

(z)

, defined as

(

)

z

n

0

n

Γ

αn

1

1

(z)

α

E

∞

∑

=

+

=

(4) Where z,α∈C;ℜ

( )

α ≥0and z >0.

In 1905, Wiman [9] extended (4) in the form

(

)

z

n

0

n

Γ

αn

β

1

(z)

β

α,

E

∑

∞

=

+

=

₍₅₎

Where

z,

α,

β

∈

C

;

ℜ

( )

α

≥

0

,

ℜ

( )

β

≥

0.

In 2000, Kiryakova [4], has studied about “Multi index M-L function”defined by

( )

₍

₎

_zn

0

n ( ₁ n ₁)... m n m

1 z

i , i 1

E ∑∞

= Γ µ + ρ Γ µ + ρ

=     

  

µ ρ _ _ _ 



 (6)

Where m>1, is an integer,

ρ

₁

,...,

ρ

m

>

0

_and

m

μ

,...,

1

μ

are arbitrary real numbers.

In 2010, Saxena and Nishimoto [7], studied an extension of M-L type function as

[

]

( )

n!

n

z

0

n

m

1

j

(

α

j

n

β

j

)

nk

γ

z

;

)

m

β

,

m

α

(

,...,

)

1

β

,

1

α

(

k

γ,

E

∑

∞

=

_∏

=

+

=

(7)

Where

m

( )

α

( )

_k

₁

_,

_j

_1,...,

_m

_and

( )

_k

_0.

1

j

;

C

γ

,

j

β

,

j

α

z,

∑

ℜ

>

ℜ

−

=

ℜ

>

=

∈

In 2012, Kalla , Haidey and Virchenko [3] , showed Multi parameter M-L type function in the following form

( )

( )

(

)

n

Λn M

λ ,...,λ

₁

1

z

HE

_{μ ,...,μ}

z

Λ

1

n 0

Γ 1 μ

_j

λ n

_j

j 1

ν

ν

_ν

+

∞

−





_{= ∑}













₌

_

_{ }

_







₊

₊



∏





=









(8)

Where

Λ

_.

1

j

λ

j

and

M

1

j

μ

j

;

1,2,...,

j

,

0

j

λ

,

C

j

μ

∑

ν

=

=

=

∑

ν

=

ν

=

>

∈

Chaurasia [2] gave Series representation of the Aleph function.

[ ]

( )

_z s

G B ! g

) s ( N , M

r , i c , i Q , i P g 1 M

1

G g 0

z N

M,

r , i c , i Q , i P

− Ω

− ∑

= ∑

∞

= =

ℵ (9)

With

G G

B

g

b

g

,

G

s

=

η

=

+

,

,

z

1

i

Q

i

P

<

<

∏ +

= =∏+ 

 



 ₊

   



 _{− −}

∑ =

∏

= ∏= 

 

  _{− −} 

  

  ₊ =

Ω

i Q

1 M j

i P

1 N j

s ji A ji a Γ s

ji B ji b 1 Γ r

1 i i

c 1

j j 1

s j A j a 1 Γ s j B j b Γ )

s ( N , M

r , i c , i Q , i P and

M N

(10)

2. MITTAG- LEFFLER TYPE E-FUNCTION

( )

( ) ( )         =           =                                                  h s , h q , h γ ,..., i s , i q , i γ ; a ρ, k r , k p , k δ ,..., 1 r , 1 p , 1 δ ; β α, | z h k E τ h 1, i s , i q , i γ ; a ρ, k 1, j r , j p , j δ ; β α, | z h k E τ z h k E τ

( )

(

α

n

β

)

Γ

k

r

n

k

p

)

k

δ

(

...

2

r

n

2

p

)

2

δ

(

1

r

n

1

p

)

1

δ

(

τ

an

z

ρn

1

h

s

n

h

q

)

h

γ

(

...

2

s

n

2

q

)

2

γ

(

1

s

n

1

q

)

1

γ

(

+





































+

−





































=

(11)

Where _{) 0}

j ( , 0 ) i ( , 0 ) ( , 0 ) ( , C j , i , , ,

z α β γ δ ∈ ℜ α ≥ ℜ β > ℜ γ > ℜ δ >

)

0

.

i

q

(

and

ℜ

≥

{ }

h

( )

or

0

i

k

1

j

p

j

r

j

i

s

i

q

,

1

,

0

;

R

,

a

,

j

r

,

i

s

;

0

j

p

_















∑

=

<

∑

=

+

ℜ

α

∈

ρ

∈

τ

≥













ℜ

( )

          ∑

= ∑= ∏= <

α −         ∏ =      α α α ℜ + = h 0 i k 1 j h 1 i 1 z 1 k 1 j j r j p j p i s i q ) i q ( when j r j p i s i q (12) k. 1,2,..., j ; h 1,2,..., i

Where = =

3. THE IMAGE OF M-L TYPE E-FUNCTION UNDER THE RIEMANN – LIOUVILLE (R-L) OPERATOR

θ

+

c

I

Theorem 3.1:

If convergence condition (3), (9) and (12) are satisfied also

θ

∈

C

_and ℜ

( )

θ

>0then the R-L transform

I

θ

_c

+

of the

Ε

-functions is

(

)

(

)

(

)

M,N

(

t c

) ( )

x

r , i c , i Q , i P c t ,..., 1 c t U ,..., 1 U V S c t h k E τ θ C I                   − ℵ       _{− −} −

+ l R Rl

( )

U R V

i i i 1

1 1 1

V A(V, R ,..., R )₁

R ,...,R 0 R ! R !

1 _{U R} 1

τ Ri _G,g 1 _{i i}

i 1

i 1 _θ

l l l _l l l ≤ ∑ = = ∑ − =

+ ∑ − η + ∑

= =















( )

(

)

i 1 i G,g

R c x G B ! g (s) N M, r , i c , i Q , i P Ω g 1 M 1

G g 0

η − ∑ = − − ∑ = ∑ ∞ = × l

(

)

(

)

(

)

              + − ∑ = + + − ∑ = + + β α − + + + × 1,a,1) G,g η 1 i i R (τ , h 1, ) i s , i ,q i (γ ; a ρ, 1,a,1) G,g η 1 i i R θ ,(τ k 1, ) j r , j p , j (δ ; , | c x 1 h 1 k E θ τ l l (13) Proof:

(

)

(

)

(

)

M,N

(

_{t c}

) ( )

_x

r , i c , i Q , i P c t ,..., 1 c t U ,..., 1 U V S c t h k E τ θ C I                   − ℵ       _{− −} −

+ l R Rl

(

)

( )(

)

(

)

(

)

( )

i i i

₍

₎

g _M,N P ,Q ,c ,r

G

1

1 1 _{( , ,..., )}

1

0 ,..., 0

1

1

M 1Ω (s)

,

! _{G 1 g 0} g! B

1

l

U R V i i

x _{an i}

x t n t c V A V R R

l l

n R R

c _l

U R i i i

R l _{t c i}

t c G g dt R i i θ τ θ η  ≤ ∑  ∞  _{− +} =

=_Γ ∫ − ∑ Φ − ∑ −

Fractional Integral Transformations of Mittag- Leffler Type E-function with Multivariable Polynomial…. / IJMA- 8(5), May-2017.

Where

( )

( )

(

αn β

)

Γ

k r n k p ) 1 δ ( ... 2 r n 2 p ) 1 δ ( 1 r n 1 p ) 1 δ (

ρn 1 h s n h q ) 1 γ ( ... 2 s n 2 q ) 2 γ ( 1 s n 1 q ) 1 γ ( n

+ 

  

  

  

     

 

−    

  

  

     

  =

Φ (14)

(

)

( )

( )

(

)

(

)

(anτ Ri ,1) 1 1

1

1 1

( , ,..., )

1 _!

,..., 0 1 0

1

1 g M, N

1Ω (s)

P , Q , c , r

M _{i i i 1}

t c g! B

G 1 g 0 _G

l G g i

l

U R V

i i _l

i

V A V R R n

l _R

l

R R i _i n

l U R

i i i

x

x t dt

c

η

θ

θ + +∑ − + − =



≤ ∑



∞

 =

= ∑ − ∏ ∑ Φ

Γ

 = = =

∑ 

 =



− _

∞ ₋



× ∑ ∑ ∫ − −



= = _{ (15)}

By using this formula

    

 

 _µ+ν−

− ν + µ Γ

ν Γ µ Γ = − ν −

∫ − µ− (b a) 1

) (

) ( ) ( dz 1 ) z b ( b

a

1 ) a z

( after simplification by the

Beta –Gamma function formula than we get the required result (13).

Special Cases:

I. R-L transform

I

θ

_c

₊

of the M-L function (6)

(

)

(

)

(

)

M,N

(

t c

) ( )

x

r , i c , i Q , i P c

t ,..., 1 c t U ,..., 1 U V S c t ) i μ ( , ) i ρ 1 ( E θ C I

    

  

     

   

− ℵ

   



 _{− −}

−

+ l R Rl

(

)

U R V

i i i 1

1 1 1

V A(V, R ,..., R )₁

R ! R !

R ,...,R 0 ₁

1 U R

Rη 1 _{i i}

i G, g _{i 1}

1 _θ

l

l

l _l

l l

i

≤ ∑

=

= ∑ −

  =

− +

∑

  ∑

 ₌  ₌

 



( )

(

)

i 1 i G,g R

c x G

B ! g

(s) N M,

r , i c , i Q , i P

Ω

g 1 M

1

G g 0

1 m

1 j

) j

μ

( 1

η − ∑ = − −

∑ =

∑ ∞

= ∏−

= Γ

×

l

(

)

(

) (

) (

)

( )

    

 

    

 

+ −

∑ =

+ −

∑ = + −

− −

×

1,1,1) g

G,

η

1

i i

R ( ; 0,1

1,1,1) g

G,

η

1

i i

R (θ , ,1 1 m

ρ

1 , 1 m

μ

,..., ,1 1

ρ

1 , 1

μ

; m

μ

, m

ρ

1 | c x 1

m E

θ

l

l

(16)

II. R-L transform

I

θ

_c

₊

of the M-L function (7)

[

]

M,N

( ) ( )

t x

r , i c , i Q , i P ) t ,..., 1 t ( U ,..., 1 U V S t ; ) m , m ( ,..., ) 1 , 1 ( k , E

θ

C I

    

  

     

   

ℵ β

α β

α γ

+ l

R R

l

(

)

1

1 1 1

R ! R ! 1

,

θ

U R V

i i i 1

V A(V, R ,..., R ) 1

R ,...,R₁ 0

U R

R _{i i}

i _{i 1}

1

l

G g

l

l

l _l

l i

η

=

− +

≤ ∑

=

− ∑

  =

∑

  ∑

  ₌

 = 



( )

_G,g

1

i i

R ) x ( G

B ! g

(s) N M,

r , i c , i Q , i P

Ω

g 1 M

1

G g 0

1 m

1 j

) j (

1 − ∑₌ −η

∑ =

∑ ∞

= ∏−

= Γ β

×

( )

(

) (

)

( )

    

 

    

 

+ − ∑ = γ

+ − ∑ = + +

×

1,1,1) g

G,

η

1 i i

R ( , k,1) , ( ; 0,1

1,1,1) g

G,

η

1 i i

R (θ , ,1 m

α

, m

β

,..., ,1 1

α

, 1

β

; 1,1 | x 2

1 m E

θ

l

l

(17)

III. R-L transform

I

_cθ₊ of the M-L function (8)

(

1

)

( )

1 λ ,...,λ μ ,...,μ

U ,..., U _{M, N}

θ 1 1

I HE ; t S ( ,..., ) ( ) x

C V P , Q , c , r

i i i

R R

l _{t t} l _t

ν ν

 _ _

ℵ

 ₊ 

  

 

(

)

1

1 1 1

R ! R !

M 1

,

θ

U R

V

i i

i 1

V

A(V, R ,..., R )

1

R ,...,R

0

1

U R

R

_{i i}

i

_{i 1}

1

l

G g

l

l

l

_l

l

i

η

=

+ − +

≤

∑

=

−

∑





=

∑





∑





₌



=





( )

g , G R ... 1 R x

G B ! g

(s) N M,

r , i c , i Q , i P Ω g 1 M

1

G g 0

1

1

j j

μ 1 Γ

1 − θ+ + + −η

∑ =

∑ ∞

= ∏−

ν

= 

    ₊

× l

(

) (

)

(

)

(

)





























Λ

+

−

∑

=

+

Λ

Λ

+

θ

+

−

∑

=

+

−

ν

−

ν

+

+

ν

+

ν

Λ

ν

×

,1)

1,

g

G,

η

1

i

i

R

(M

;

1,

,1)

1,

g

G,

η

1

i

i

R

(M

,

,1

1

λ

,

1

μ

1

,...,

,1

1

λ

,

1

μ

1

,...,

μ

,1

λ

|

x

1

E

M

l

l

(18)

IV. If we Substitute Multivariable Polynomial

(

)

l l _,...,x

1 x U ,..., 1 U V

S and Aleph function is unity, then we get the results reduce in [1].

4. THE IMAGE OF M-L TYPE E-FUNCTION UNDER ERDELYI- KOBER (E-K) OPERATOR

Ξ

η

₊

,

θ

0

Theorem 4.1: If convergence condition (3), (9) and (12) are satisfied also

η

,

θ

∈

C

,

ℜ

(

η

).

>

0

and ℜ

( )

θ

>0

then the E-K transform

Ξ

η₀₊,θ of the E-function is

( )

t 1,...,t M,N

( )

t

( )

x

r , i c , i Q , i P U

,..., 1 U V S t h k E

τ θ η,

0

Ξ

    

  

     

   

    

  

ℵ

+ l

R R

l

(

)

1 1 1

R !₁ R !

τ _, 1

U R_{i i} V i 1

V A(V, R ,..., R )₁ R ,...,R 0

1 U R

R _{i i}

i _{i 1}

1

l G g

l

l

l _l

l i

θ η

η

=

+ + − +

≤ ∑

=

− ∑

  =

∑

  ∑

  ₌

 = 



( )

_G,g

1

i i

R x

G B ! g

(s) N M,

r , i c , i Q , i P Ω g 1 M

1

G g 0

η − ∑ = −

∑ =

∑ ∞

= ×

l

(

)

(

)

    

 

    

 

+ −

∑ = + θ +

+ −

∑ = + η + + β

α +

+ ×

1,a,1) G,g

η 1

i i

R (τ

, h 1, ) i s , i ,q i (γ ; a ρ,

1,a,1) G,g

η 1

i i

R θ

,(τ k 1, ) j r , j p , j (δ ; , | x 1 h

1 k E τ

l

l

(19)

Proof:

We get the E-K transform

Ξ

η

₊

,

θ

0

of the E-function as follows

( )

t 1,...,t M,N

( )

t (x)

r , i c , i Q , i P U

,..., 1 U V S t h k E

τ θ η,

0

Ξ

    

  

     

   

    

  

ℵ

+ l

R R

Fractional Integral Transformations of Mittag- Leffler Type E-function with Multivariable Polynomial…. / IJMA- 8(5), May-2017.

(

)

( )

( )

( )

i i i

g M,N P ,Q ,c ,r

G

U R V

i i ₁

i 1 1

V A(V, R ,..., R )

1

( ) _{0 R ,..., R 0} ! !

0 ₁ 1

U R i i i 1

M 1Ω (s) _,

g! B

G 1 g 0

l

R R

x l

x an t t

x t t n t

l R R

l

n _l l

G g

t dt

η θ _{η θ τ}

η

η



≤

 ∑

− − _∞

 _{− +} =

= _Γ ∫ − ∑ Φ ∑ −

 = =

 ∑

 =

− 

∞ −



× ∑ ∑



= = _



(20)

( )

(

)

( )

(

)

i ,

1

( R 1) 1

1 0

U R V

i i ₁

i 1

V A(V, R ,..., R )

1

( ) _{0 R ,..., R 0} ! !

1

1 _{U R}

i i

i 1

g M, N

1Ω (s)

P , Q , c , r M

i i i

g! B

G 1 g 0 _G

l G g i

x an

l

R

R _l

x t t

n

l R R

l

n _l l

x t t dt

θ τ η η

η θ η

=

+ + + − + − −



≤

 _{− −} ∑

∞

 =

= ∑ Φ ∑ −

Γ

 = =

 ∑

 =



− _

∞ ∑



× ∑ ∑ −



= = _



∫



( )

( )

( )

dt g , G 1 i i

R an

x , 1 g , G 1 i i

R an

G B ! g

(s) N M,

r , i c , i Q , i P

Ω

g 1 M

1 G g 0

! R t ! 1 R

1 t 0

n

) R ,..., 1 R V, ( A 1

i

V i R i U

0 R ,..., 1 R

1

i i

R i U V n

) ( x

    

  η − ∑ = + τ + + θ + η

    

  

η + η − ∑ = + τ + + θ β −

∑

= ∑

∞

= ×

     

 

∑ ∞

= ∑

∑

= ≤

=

∑ = − Φ

η Γ

θ − η − =

l l

l l R R l l

l l



(21)

By using the Beta –Gamma function, we get the result (19).

Special Cases:

I. E-K transform

Ξ

η₀₊,θ of the M-L type function (6)

( )

M,N

( ) ( )

t x

r , i c , i Q , i P ) t ,..., 1 t ( U ,..., 1 U V S t ) i μ ( , ) i ρ 1 ( E

θ η,

0

Ξ

    

  

     

   

ℵ

+ l

R R

l

(

)

U R V

i i i 1

1 1 1

V A(V, R ,..., R )₁

R ! R !

R ,...,R₁ 0 ₁

R_{i ,} 1 U R_{i i}

1 _{i 1}

l

l

l _l

l l

G g i

θ

η

η

≤ ∑ =

= ∑ −

 _{+ ∑ − +}  =

  ∑

=

  =



( )

_G,g

1

i i

R x G

B ! g

(s) N M,

r , i c , i Q , i P

Ω

g 1 M

1

G g 0

1 m

1 j

) j

μ

(

1 − ∑₌ −η

∑ =

∑ ∞

= ∏−

= Γ

×

l

(

) (

) (

)

( )





























+

−

∑

=

+

θ

+

−

∑

=

+

+

η

−

−

×

1,1,1)

g

G,

η

1

i

i

R

(

;

0,1

1,1,1)

g

G,

η

1

i

i

R

θ

(

,

,1

1

m

ρ

1

,

1

m

μ

,...,

,1

1

ρ

1

,

1

μ

;

m

μ

,

m

ρ

1

|

x

1

m

E

0

l

l

(22)

II. E-K transform

Ξ

η

₊

,

θ

0

of the M-L type function (7)

[

]

M,N

( ) ( )

t x

r , i c , i Q , i P ) t ,..., 1 t ( U ,..., 1 U V S t ; ) m , m ( ,..., ) 1 , 1 ( k , E

θ η,

0

Ξ

    

  

     

   

ℵ β

α β

α γ

+ l

R R

(

)

U R V

i i i 1

1 1 1

V A(V, R ,..., R )₁

R ! R !

R ,...,R 0 ₁

1 U R

R _, 1 _{i i}

i _{i 1}

1

l

l

l _l

l l

G g i

θ η

η

≤ ∑

=

= ∑ −

  =

+ ∑ − +

  ∑

 ₌  ₌

 



( )

_G,g

1

i i

R

x

G B ! g

(s) N M,

r , i c , i Q , i P

Ω

g 1 M

1

G g 0

1 m

1 j

) j (

1 − ∑₌ −η

∑ =

∑ ∞

= ∏−

= Γ β

×

l

( )

(

) (

)

( )

    

 

    

 

+ − ∑ = + θ γ

+ − ∑ = + η + +

×

1,1,1) g

G,

η

1 i i

R (

, k,1) , ( ; 0,1

1,1,1) g

G,

η

1 i i

R (θ

, ,1 m

α

, m

β

,..., ,1 1

α

, 1

β

; 1,1 | x 2

1 m E 0

l

l ₍₂₃₎

III. E-K transform

Ξ

η₀₊,θ of the M-L type function (8)

(

1

)

( )

1 λ ,...,λ μ ,...,μ

U ,..., U

η,θ 1 1 M, N

Ξ HE ; t S ( ,..., ) ( ) x

0 V P , Q , c , r

i i i

R R

l _{t t} l _t

ν ν

 _ _

ℵ

 ₊  

  

 

(

)

1 1 1

R !₁ R !

M _, 1

U R

V

i i

i 1

V

A(V, R ,..., R )

₁

R ,...,R

0

1

U R

R

_{i i}

i

_{i 1}

1

l G g

l

l

l

_l

l

i

θ η

η

=

+ + − +

≤

∑

=

−

∑





₌

∑





∑





₌



=





( )

g , G R ... 1 R x

G B ! g

(s) N M,

r , i c , i Q , i P Ω g 1 M

1

G g 0

1

1

j j

μ 1 Γ

1 _∑ − + + −η

= ∑ ∞

= ∏−

ν

= 

    ₊

× l

(

) (

)

(

)

(

)

    

 

    

 

Λ + − ∑ = + θ + γ

Λ

Λ + θ + − ∑

= + η + −

ν − ν + +

ν + ν Λ ν ×

,1) 1, g G, η 1

i i

R (M

, k,1) , ( ; 1,

,1) 1, g G, η 1

i i

R (M

, ,1 1 λ , 1 μ 1 ,..., ,1 1 λ , 1 μ 1 ,..., μ ,1 λ | x 1 E M

l

l

(24)

IV. If we Substitute Multivariable Polynomial

(

)

l l _,...,x

1 x U ,..., 1 U V

S and Aleph function is unity, then we get the results reduce in [1].

5. THE IMAGE OF THE M-L TYPE E- FUNCTION UNDER A GENERALIZED INTEGRAL OPERATOR. Theorem 5.1: If convergence condition (3), (9) and (12) are fulfilled also

η

,

θ

,

σ

∈C,ℜ(

η

).>0,ℜ(

σ

)>0,ℜ

( )

θ

>0

and

t

,

x

,

υ

∈

R

then

(

)

(

)

{

(

)

}

(

)

(

)

M,N _,r

(

s t

)

ds

i ,c i ,Q i P i t s ,..., 1 t s U ,..., 1 U V S σ t s h k E τ x

t

1 θ t s 1 η s

x ν − _ − − _ℵ −

∫ − − − − l R R

(

)

( )

g M, N

U R V _{1Ω (s)}

i i _{M P , Q , c , r}

i 1 1 1 _{i i i}

V A(V, R ,..., R )

1 _{R ! R ! g! B}

R ,...,R 0 _{1 G 1 g 0 G}

1 U R_{i i}

i 1

l

l

l _l

l ≤

∑ _∞ −

=

= ∑ − ∑ ∑

= _{= =}

∑ =



1 g , G 1

i i

R )

( ) , g , G

(

x

t

1

i

i

R

− η − ∑ = + θ + η η

τ σ + η − +

θ β

×

∑

−

=

l

Fractional Integral Transformations of Mittag- Leffler Type E-function with Multivariable Polynomial…. / IJMA- 8(5), May-2017.

(

)

(

)

(

)

    

 

    

 

σ −

∑ = + τ σ + θ

σ −

∑ = + τ σ + θ + η β

α σ − ν + + ×

,1) ,a G,g η 1

i i

R (

, h 1, ) i s , i ,q i (γ ; a ρ,

,1) ,a G,g η 1

i i

R ,(

k 1, ) j r , j p , j (δ ; , | t x 1 h

1 k E τ

l

l

(25)

Proof: The theorem is proved as follows

(

)

(

)

{

(

)

}

(

)

(

)

M,N

(

s t

)

ds

r , i c , i Q , i P t

s ,..., 1 t s U ,..., 1 U V S σ t s h k E τ x

t

1 θ t s 1 η s

x ℵ −

  



 _{− −}

− ν

∫ − − − − l R Rl

(

)

(

)

( ) (

{

)

}

(

)

( )

i i i

g _M,N P ,Q ,c ,r

G

U R V i i

x _{η 1 θ 1 σ} anτ _{i 1}

x s s t s t V A(V, R ,..., R )

1

0 R ,..., R 0

t ₁

U R i i i 1

M 1Ω (s)

( ) _{( )} ,

! _{G 1 g 0} g! B

1

l

n

l l

n

l

R_i

l _{s t G g}

s t ds

R

i i

ν

η 

≤

 ∑

+ ∞

 _{− −} =

=∫ − − ∑ Φ − ∑ −

 = =

 ∑

 =

− 

∞ −

− _

× ∏ ∑ ∑ −



= =

= _

( )

(

)

( )

g M,N_{i i i}

₍

₎

₍

₎

P ,Q ,c ,r

G

U R V i i

i 1 1 1

anτ _{V A(V, R ,..., R )}

1 _{! !}

0 R ,..., R 0 ₁

1 _{U R}

i i i 1

R 1

i ,

x

M 1Ω (s) _{η 1} 1

x s s t

g! B

G 1 g 0 t

l

n

l R R

l

n _l

l

l

an

G g i

ds

ν

θ σ στ η



≤

 ∑

∞

 ₊ =

= ∑ Φ ∑ −

 = =

 ∑

 =



+ ∑ + + − − − 

∞ − ₋ = 

× ∑ ∑ ∫ − − 



= =

  

(26)

By using this formula

    

 

 _µ+ν−

− ν + µ Γ

ν Γ µ Γ = − ν −

∫ − µ− (b a) 1

) (

) ( ) ( dz 1 ) z b ( b

a

1 ) a z

( after simplification by the

Beta –Gamma function formula than we get the required result (25).

Corollary 5.2: If convergence conditions (9) and (3) are satisfied also

η

,

θ

,

σ

∈

C

,

ℜ

(

η

).

>

0

,

ℜ

(

σ

)

>

0

,

ℜ

( )

θ

>

0

and

0

t

,

R

,

x

υ

∈

=

in theorem (5.1) then

(

)

{

}

M,N

( )

s ds

r , i ,c i ,Q i P ) s ,..., 1 s ( U ,..., 1 U V S σ s h k E τ x

0

1 θ s 1 η s

x ν ℵ

∫ − − − l R Rl

(

)

( )

g M,N

U R V _{1Ω (s)}

i i _{M P ,Q ,c ,r}

i 1 1 1 _{i i i}

V A(V, R ,..., R )

1 _{R ! R ! g! B}

R ,...,R 0 _{1 G 1 g 0 G}

1 U R

i i i 1

l

l

l _l

l ≤

∑ _∞ −

=

= ∑ − ∑ ∑

= _{= =}

∑ =



1 g , G 1

i i

R x

) , g , G (

1

i

i

R

− η − ∑ = + θ + η η τ σ + η − +

θ β

×

∑

=

l

l

(

)

( )





























σ

−

∑

=

+

τ

σ

+

θ

σ

−

∑

=

+

τ

σ

+

θ

+

η

β

α

σ

ν

+

+

×

,1)

,a

G,g

η

1

i

i

R

(

,

h

1,

)

i

s

,

i

,q

i

(

γ

;

a

ρ,

,1)

,a

G,g

η

1

i

i

R

,(

k

1,

)

j

r

,

j

p

,

j

(

δ

;

,

|

x

1

h

1

k

E

τ

l

l

Corollary 5.3: If convergence conditions (9) and (3) are satisfied also

θ

,

σ

∈

C

,

,

ℜ

(

σ

)

>

0

,

ℜ

( )

θ

>

0

and

,

1

,

0

t

,

R

,

x

υ

∈

=

η

=

in theorem (5.1) then

{

}

M,N

( )

s ds

r , i ,c i ,Q i P ) s ,..., 1 s ( U ,..., 1 U V S σ s h k E τ x

0 1 θ

s ν ℵ

∫ − l R Rl

(

)

( )

g M, N

U R V _{1Ω (s)}

i i _{M P , Q , c , r}

i 1 1 1 _{i i i}

V A(V, R ,..., R )

1 _{R ! R ! g! B}

R ,...,R 0 _{1 G 1 g 0 G}

1 U R_{i i}

i 1

l

l

l _l

l ≤

∑ _∞ −

=

= ∑ − ∑ ∑

= _{= =}

∑ =



(

)

( )

    

 

    

 

σ −

∑ = + τ σ + θ

σ −

∑ = + τ σ + θ + η β

α σ ν + +     

 

    

 

η − ∑ = + θ + τ σ

η − ∑ = + θ

×

,1) a , g G, η 1

i i

R (

, h 1, ) i s , i q , i (γ ; a ρ,

,1) a , g G, η 1

i i

R (

, k 1, ) j r , j p , j (δ ; , | t 1 h

1 k E τ

g , G 1

i i

R g , G 1

i i

R x

l

l

l l

(28)

Special Cases:

I. General integral transform of the M-L type function (6)

(

) (

)

{

(

)

}

(

)

(

)

M,N

(

s t

)

ds

r , i ,c i ,Q i P t

s ,..., 1 t s U ,..., 1 U V S σ t s i μ , i ρ 1 E x

t

1 θ t s 1 η s

x ℵ −

  



 _{− −}

− ν

∫ − − − −

          

 l

R R

l

(

)

( )

g

M, N

U R

V

₁

_Ω

_(s)

i i

_M

_{P , Q , c , r}

i 1

1

1

_i

_i

_i

V

A(V, R ,..., R )

₁

R !

R !

g! B

R ,...,R

₁

0

₁

_G

_{1 g}

₀

_G

U R

_{i i}

i 1

l

l

l

_l

l

≤

∑

_∞

−

=

=

∑

−

∑

∑

=

₌

₌

∑

=



1 g , G 1

i i

R )

( 1

m

1 j

) j μ (

) , g , G 1

i i

R (

t

x

− η − ∑ = + θ + η

∏− = Γ

η η − ∑ = + θ β

×

−

l l

(

)

( )

    

 

    

 

σ −

∑ = + θ

σ −

∑ = + θ + η − −

σ − ν ×

,1) , g G, η 1

i i

R (

, 0,1

,1) , g G, η 1

i i

R (

, ) ,1 1 m ρ 1 , 1 m μ ( ,..., ) ,1 1 ρ 1 , 1 (μ ; ) m μ , m ρ 1 ( t x 1

m E 0

l

l

(29)

II. General integral transform of the M-L type function (7)

(

) ( )

{

(

)

}

( )

( )

M,N _,r

( )

s tds

i c , i Q , i P t s ,..., 1 t s ,...,U 1 U V S t s ; m β , m α ,..., ) 1 β , 1 α ( k γ, E x

t

1 θ t s 1 η s

x − _ − − _ℵ −

∫ − − − − l R Rl

(

)

( )

g M, N

U R V _{1Ω (s)}

i i _{M P , Q , c , r}

i 1 1 1 _{i i i}

V A(V, R ,..., R )

1 _{R ! R ! g! B}

R ,...,R 0 _{1 G 1 g 0 G}

1 U R

i i i 1

l

l

l _l

l ≤

∑ _∞ −

=

= ∑ − ∑ ∑

= _{= =}

∑ =



1 g , G 1

i i

R

) ( m

1 j

) j (

) , g , G 1

i i

R (

t

x

− η − ∑ = + θ + η

∏ = Γ β

η η − ∑ = + θ β

×

−

l l

(

)

( )

( )

    

 

    

 

− ∑ = + θ

− ∑ = + θ + η −

+ ×

,1) 1 , g G, η 1

i i

R (

, ) k,1 γ, ( , 0,1

,1) 1 , g G, η 1

i i

R (

, ) ,1 m α , m β ( ,..., ) ,1 1 α , 1 β ( ; 1,1 t x 2

1 m E 0

l

l

Fractional Integral Transformations of Mittag- Leffler Type E-function with Multivariable Polynomial…. / IJMA- 8(5), May-2017.

III. General integral transform of the M-L type function (8)

(

) (

)

{

(

)

}

(

)

(

)

M,N

(

s t

)

ds

r , i c , i Q , i P t

s ,..., 1 t s ,...,U 1 U V S σ t s λ ,..., 1 λ

μ ,..., 1 μ HE x

t

1 θ t s 1 η s

x ℵ −

  



 _{− −}

− ν ν ν

∫ − − − − l R Rl

(

)

( )

g M, N

U R_{i i} V 1Ω (s)

P , Q , c , r M

i 1 1 1 _{i i i}

V A(V, R ,..., R )₁

R ! R ! g! B

R ,...,R₁ 0 _{1 G 1 g 0 G}

U R_{i i} i 1

l

l

l _l

l ≤

∑ _∞ −

=

= ∑ − ∑ ∑

= _{= =}

∑ =



1 g , G 1

i i

R )

( 1

-1 j

) j 1 (

) , g , G 1

i i

R M

(

t

x

− η − ∑ = + θ + η

∏ ν

= Γ +µ

η η − ∑ = + σ + θ β

×

−

l l

(

) (

)

(

)

( )





























Λ

σ

−

∑

=

+

θ

+

σ

γ

Λ

Λ

σ

θ

+

−

∑

=

+

η

+

σ

−

ν

−

ν

+

+

ν

+

ν

Λ

σ

−

ν

ν

×

,1)

,

g

G,

η

1

i

i

R

M

(

,

k,1)

,

(

;

1,

,1)

,

g

G,

η

1

i

i

R

M

(

,

,1

1

λ

,

1

μ

1

,...,

,1

1

λ

,

1

μ

1

,...,

μ

,1

λ

|

)

t

x

(

1

E

M

l

l

(31)

IV. If we Substitute Multivariable Polynomial

(

)

l l _,...,x

1 x U ,..., 1 U V

S and Aleph function is unity, then we get the results reduce in [1].

REFERENCES

1. Bhatter S. and Faisal S.M., A family of Mittag- Leffler type functions and its relation with basic special functions. Int. J. of Pure Appl. Math. Volume 101 No. 3 2015, 369-379

2. Chaurasia V.B.L and Singh Y. New generalization of integral equations of fredholm type using Aleph-function Int. J. of Modern Math. Sci. 9(3), 2014, p 208-220.

3. Kalla S.L., Haidey V., and Virchenko N.O., A generalized multiparameter function of Mittag-Leffler type. Integral Transforms Special Functions 23(12), (2012), 901-911.

4. Kiryakova V.S., Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus. J.Comp. Appl. Math., 118(1-2), 241-259.

5. Mittag-Leffler G.M., Sur lanouvelle function Eα(x),C.R. Acad. Sci. Paris, 137, (1903), 554-558.

6. Samko S.G. Kilbas A.A. and Marichev O.I., Fractional Integrals and Derivatives and Some of Their Applications (in Russian), Nauka I Tekhnica, Minsk 1987 (English translation: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Reading, 1993).

7. Saxena R.K. and Nishimoto K., N-fractional Calculus of generalized Mittag- Leffler functions, J.Fract. Calc. 37, (2010), 43-52.

8. Srivastava H.M., and Garg M., Some integral involving a general class of polynomials and the multivariable H-function, Rev. Romaine Phys. 32, 685-692, (1987).

9. Wiman A., Uber den fundamental Satz in der theorie der Funktionen ¨ E_α(x), Acta Math., 29 (1905), 191-201.

Source of support: Nil, Conflict of interest: None Declared.