G LOBAL J OURNAL OF E NGINEERING S CIENCE AND R ESEARCHES
INTEGRAL TRANSFORMS OF (3 M)-PARAMETRIC MULTI-INDEX MITTAG- LEFFLER FUNCTION
Jitendra Daiya*1and Jeta Ram2
*1,2
Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur
ABSTRACTThe object of this paper is to evaluate the Euler, Laplace, Whittaker, K- transform and fractional Fourier transform of order α, 0 < α ≤ 1, of the (3m)-parametric multi-index Mittag-Leffler function defined by Paneva-Konovska [6].
The results established in this paper would provide extensions of those given in earlier works. The result obtained is useful applied problem of science and engineering.
Mathematics Subject Classification: 33E12, 33C40
Keywords- Parametric multi-index Mittag-Leffler function Euler transform, Laplace transform, Whittaker transform, k- transform, fractional Fourier transform, generalized wright function.
I. INTRODUCTION
The (3m)-parametric multi-index Mittag-Leffler function introduced by Paneva-Konovska [6] is defined as
( ), ( ),( )
0
( ) ( )
( ) ( ) ( !)
i i
i i i i
m k k
k m
E z z
k k
1
1 1
0
( ) . . . ( ) ( ) ( ). . . ( ) ( !)
m
m m
k k k
k m
z
k k k
(1); , ,
i i i; Re( ) 0,
i1, 1, 2, . . . . k R C m i m
where
( )
k is the pochhammer symbol1 ; ( 0; \{0})
( )
( )
k( ) ( 1) . . . ( 1) ; ( ; )
k C
k
k k n N C
Particular cases
(i) For
m 1,
equation (1) reduces to the generalized Mittag-Leffler function [7],1
, ,
0
( ) ( )
( ) ( )
( ) !
k k
k
E z z E z
k k
(2)(ii) For
1,
equation (2) reduces to the Mittag-Leffler function [5]1, , 0
(1) ( )
( ) ( )
( ) !
k k
k
E z z E z
k k
(3)(iii) For
1,
equation (3) reduces to the Mittag-Leffler function [4],1 0
(1) ( )
( ) ( )
( 1) !
k k
k
E z z E z
k k
(4)More detail about Mittag-Leffler function and their application can be found in the paper by Saxena et al [9, 10, and 11].
The following definitions are also needed in the analysis that follows:
Definition 1 Euler Transform
The Euler transform of the function
f z ( )
is defined as1 1 1
0
{ ( ) ; , }
a(1 )
b( ) , , Re ( ) 0, Re ( ) 0
B f z a b z
z
f z dz a b C a b
(5)Definition 2:
Laplace Transform
The Laplace transform of the function
f t ( )
, denoted by F (s) is defined by the equation0
( ) ( ) ( ) { ( ) ; }
st( ) Re ( ) 0
F s Lf s L f t s e f t dt s
(6)which may be symbolically written as
( ) { ( ) ; } ( )
1( ) ; F s L f t s or f t L
F s t
provided that the function f t
( )
is continuous fort 0
and if exponential order ast .
Definition 3 Let
u u t ( )
be a function of the spaceS R ( )
, the Schwartzian space of the function that decay rapidly at infinity together with all derivatives.The Fourier transform is given by the integral
ˆ ( ) [ ] ( ) ( ) exp ( )
u u
U t i t dt
(7)and the inverse Fourier transform can be defined as
1
ˆ ( ) 1 ˆ ( ) exp ( )
2
Ru t u i t d
(8)Definition 4 (Lizorkin space) : LetV R
( )
be the set of functions
( )
( ) ( ) :
n(0) 0, 0,1,2,... .
V R S R n
(9)The Lizorkin space of function
( )
R is defined as
( ) R S R ( ) : [ ] V R ( ) .
(10)Definition 5: Let u be a function belonging to
( ). R
The Fractional Fourier transform of the order
, 0 1
is defined byˆ ( ) [ ]( )
i 1/ t( )
u w u
Re
u t dt
(11)If put
1,
equation (11) reduces to the conventional Fourier transform and for 0,
it reduces to the Fractional Fourier Transform defined by Luchko et al [2].Lemma 1 Let u be a function of the space
( ), R
let
be a real number,0 1,
then[ ]( ) u [ ]( ), u x for x 1/
(12)The inverse Fractional Fourier transform of the order
, 0 1, u ( ) R
is defined as
1/ 11
ˆ ( ) ( ) 1 ˆ ( )
2
i t
u t
Re u dw
(13)The following result will be required in evaluating the integral (22).
1/2 1 , 0
(1/ 2 ) (1/ 2 )
( ) Re( ) 1/ 2.
(1 )
e t
W
t dt
(14)where the Whittaker function
W
,( ) z
is defined in [1](see also Mathai et al [3]), , ,
( 2 ) (2 )
1 1
2 2
( ) ( ) ( )
W
z M
z M
z
(15)
where
M
,( ) z
is defined as1/2 1/2
, 1 1
1
( ) ; 2 1 ; .
2
M
z z
e
zF z
Mathai et al [3, p. 54, Eq. 2.37] defined result will be used in evaluating the integral (27).
1 2
0
( ) 2 Re( ) 0.
t K a x dx a 2 a
(16)The generalized Wright hypergeometric function p
q z
is defined by Wright [13-15] (also see [12, p 21]) in the following form:
1 1
1 1
, ,..., , , ,..., , ;
p p
q q
p q
a A a A
b B b B Z
Γ Γ a b A B n n z n
n!
0
n q
1
j j j
p 1
i i i
(17)
where
a
i, b
j C
andA
i, B
j R i 1 , .. .. , p ; j 1 , . .. , q
and the defining series (17) converges for. p 1
1 i A i q
1
j B j
Definition 6: Let
i 0,
i,
i C ;
i 0, 1, 2, . . . for i 1, 2,..., . m
Then the multi-index Mittag- Leffler functions (1) are Wright’s generalized hypergeometric functions of the form given by [6, theorem 3, p 1093]
1 1
1
( ), 1 ( ),( ) 2 1
1
, 1 , . . . , , 1 ( ) ;
, , . . . , , , (1,1), . . . ,(1,1)
i i i i
m m
m m
m
m m
i
E z z
(18)Theorem 1. (Euler Transform): If
k R m ; 1;
i, , , ,
i i C ;Re( ) 0,
i i 1,2,..., , m
then1
1 1 ( ),
( ),( ) 0
(1 )
ii i
z
z
E
mxz d z
1 1
1
1 2
1
, 1 ,. . . , 1 , ,
( ) ;
, ,. . . ,
m,
m, (1,1),. . . ,(1,1),( , )
i
m
m m
m
i
x
(19)Proof : - Using equation (1) and (5), it gives
1
1 1 ( ),
( ),( ) 0
(1 )
ii i
z z E
m xz dz
1 1 1
0 0
( ) ( )
(1 )
( ) ( !)
i
i i
k k k m
z z xz dz
k k
1 1 1
0 0
( ) ( )
(1 ) ( ) ( !)
i
i i
k k
k k m
x z z dz
k k
0
( ) ( ) ( ) ( )
( ) ( !) ( )
i
i i
k k k m
x k
k k k
01
( ) ( )
( ) ( )
( ) ( ) ( !)
i
i i
i
k
m m
k i
k k x
k k k
This completes the proof of the Theorem 1.
Corollary 1.1 For
m 1,
equation (19) reduces in the following form given by Saxena [8]
1 1 1
2 2 0 ,
, 1 , ,
( ) ;
(1 )
, ,( , )
z z E xz dz x
(20)Theorem 2. (LaplaceTransform): If
k R m ; 1;
i, , , ,
i i C ; Re ( ) 0, Re ( ) 0, s
i
1, 2,..., x 1,
i m and
s
then
1 ( ),
( ),( ) 0
i
i i
sz m
z e E x z dz
1
1 1
1 2 1
1
, 1 ,. . . , 1 , ,
, ,. . . ,
m,
m(1,1),. . . ,(1,1) ;
i
m
m m
m
i
s x
s
(21)
Proof : - Using equation (1) and (6) and gamma function formula, we obtain
1 ( ),
( ),( ) 0
i
i i
sz m
z e E xz dz
1
0 0
( ) ( )
( ) ( !)
i
i i
sz k k
k m
z e xz dz
k k
1
0 0
( ) ( ) ( ) ( !)
i
i i
k k sz
k k m
x z e dz
k k
0
( ) ( ) [ ]
( ) ( !)
i
i i
k k
m k
n
x k
k k s
0
1
( ) [ ] ( )
( ) ( !)
i
i i
i
k
m k m
n i
k k
s x
k s k
This completes the proof of the Theorem 2.
Corollary 2.1 For
m 1
equation (21) reduces in the following form given by Saxena [8]
1 ,
, 2 1 0
, 1 , , , ;
sz
a x
z e E x z dz s
s
(22)Theorem 3. (Whittaker Transform): If
k R m ; 1;
i, , , ,
i i C ;Re( ) 0,
i i 1,2,..., , m Re ( ) 0, Re ( ) 1 / 2 then
1 /2 ( ),
, ( ),( )
0
( )
ii i
pt m
t e W pt E wt dt
1
1 1
2 2
1
, 1 ,. . . , 1 , 1/ 2 ,
, ,. . . ,
m,
m,(1,1),. . . ,(1,1), 1 , ;
i
m
m m
m
i
p w
p
(23)
1 /2 ( ),
, ( ),( )
0
( )
ii i
pt m
t e W pt E wt dt
1 /2
, 0 0
( ) ( ) ( )
( ) ( ) ( !)
i
i i
k k
pt k
k m
w t
t e W pt dt
k k
It we set
pt
, then above line is equal to1
/ 2 ,
0 0
( ) ( ) 1
( ) ( ) ( !)
i
i i
k k k k m
e W w d
p k k p p
/ 2 1
,
0 0
( ) ( ) ( )
( ) ( !)
i
i i
k k k
k m
p w e W d
k k p
0
( ) (1/ 2 ) (1/ 2 )
(1 )
( ) ( !)
i
i i
k k
k m
k k w
p k k k p
01
( ) (1/ 2 ) (1/ 2 )
(1 )
( ) ( !)
i
i i
i
k
m m
k i
p k k k w
k k k p
This completes the proof of the Theorem 3.
Corollary 3.1 For
m 1,
equation (23) reduces in the following form given by Saxen [8]
1 /2
, , 3 2
0
, 1 , 1/ 2 , ( ) ;
, , 1 ,
pt
p w
t e W pt E wt dt p
(24)Fractional Fourier Transform (FFT) of multi-index (3m-Parametric)Mittag-Leffler function
Theorem 4: If
k R m ; 1;
i, ,
i i C ;Re( ) 0,
i i 1,2,..., , 0 m 1,
for FFT of order
of the multi-index (3m parametric) Mittag-Leffler function
1 1 ( 1)/
( ), ( ),( )
1 0
( ) ( ) ( 1) ( 1)
( ) ( ) ( !)
i i
i i i
i i
k k k
m m
k m i
k i k
E t
k k
(25)Proof : - Using equation (7) and (11) and gamma function formula, it gives
( ),
(
i),( ) ( )
i i
E
mt
1/
( ),
( ),( )
i( )
i i
i t m
R e
E t dt
1 /
0
( ) ( )
( ) ( !)
i
i i
i t k k
R m
k
e t d t
k k
1 /
0
( )
( ) ( !)
i
i i
i t k
k
m R
k
e t d t
k k
If we set
i 1/ t
, then 01/ 1/
0
( )
( ) ( !)
i
i i
k
k m
k
e d
k k i i
1 ( 1) /
0 0
( )
( ) ( ) ( 1) ( !)
i
i i
k k
k k k m
k
e d
k i k
1 ( 1) / 0
( ) ( 1)
( ) ( ) ( 1) ( !)
i
i i
k
k k k m
k
k
k i k
1 ( 1) /
0
( ) ( ) ( 1) ( 1)
( ) ( !)
i
i i
k k k
k k m
i k
k k
This completes the proof of the Theorem 4.
Corollary 4.1 For
m 1,
equation (25) reduces in the following form
1 ( 1)/( ) ( ),( )
0
1 ( ) ( ) ( 1)
( ) ( ) ( )
k k k
k
E t k i
k
(26)Theorem 5. (K-Transform):
k R m ; 1; , , , ,
i i i C ;Re( ) 0,
i i 1,2,..., , m
then
1 ( ), 2
( ),( ) 0
( )
ii i
t K at E
mxt dt
1
1 1
2 2
2 2 1 2
1
, 1 ,. . . , 1 , ,
2 2 ; 2
, ,. . . , , ,(1,1),. . . ,(1,1),
i m m
m n
m m
m n
i
p
a x
a
(27)
Proof : - Using equation (1) and (16) , we obtain
1 ( ), 2
( ),( ) 0
( )
ii i
t K at E
mxt dt
1 2
0 0
( ) ( ) ( )
( ) ( ) ( !)
i
i i
k k
k k m
x t
t K at dt
k k
2 1
0 0
( ) ( ) ( )( !) ( )
i
i i
k k
k k m
x t K at dt
k k
2 2 ( 2 )
0
( ) ( ) 2
2 2
( )( !)
i
i i
k k k
k k m
x k
k k a
2 2
0
( ) ( ) 2 2
2 ( )( !) 2
i
i i
k k
k m
k
x k
a k k a
This completes the proof of the Theorem 5.
Corollary 5.3 For
m 1,
equation (27) reduces in the following form given by Saxena [8]
1 2 2 2
3 1
, 2
0
,1 , ,
2 2
( ) ; 2
, ,
n n
a x
t K at E x t dt
a
(28)II. ACKNOWLEDGEMENT
This work is supported by Post-Doctoral Fellowship of the National Board of Higher Mathematics (NBHM) India to author (Jitendra Daiya).
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