Vol. 9, No. 1, 2015, 45-51
ISSN: 2279-087X (P), 2279-0888(online) Published on 1 January 2015
www.researchmathsci.org
Certain Properties of Extended Wright Generalized
Hypergeometric Function
S. C. Sharma1 and Menu Devi2 1
Department of Mathematics, University of Rajasthan
Jaipur- 302004, Rajasthan, India. E-mail: [email protected]
2
Department of Mathematics, University of Rajasthan
Jaipur- 302004, Rajasthan, India. E-mail: [email protected]
Received 17 November 2014; accepted 4 December 2014
Abstract. In this paper, we obtain the extended Wright generalized Hypergeometric
function using extended Beta function. We also obtain certain integral representations, Mellin transform and some derivative properties of extended Wright generalized Hypergeometric function. Further, we represent extended Wright generalized Hypergeometric function in the form of Laguerre polynomials and Whittaker function.
Keywords: Extended Wright generalized Hypergeometric function; Extended Beta
function; Laguerre polynomials; Whittaker function; Mellin transform; Fractional derivative operator
AMS Mathematics Subject Classification (2010): 33C70, 33B15, 33C45, 33C15,
44A05, 26A33
1. Introduction
The Wright function was introduced and investigated by British mathematician E. M. Wright in the form of power series [3-6]:
0 1
0
____ 1
( , ; )
( , ) ( ) !
k
k
z
z z
k k
φ α β ψ
α β α β
∞
=
= =
Γ +
∑
; β> −1, α, z∈C (1)Fox and Wright investigated the more general function termed as Wright generalized Hypergeometric function given as [1-2]:
1, 1,
( , ) 1
( , )
0 1
( )
; !
( )
i i m j j n
m
i i k
a i
m n b n
k
j j j
a k
z z
k
b k
α β
α ψ
β
∞ =
=
=
Γ +
=
Γ +
∏
∑
∏
z a b, ,i j∈C;α βi, j R + ∈
(2) (i=1,..., ;m j=1,..., )n
where
1 1
1
n m
j i j i
β α
= =
− > −
∑
∑
, for large values of z.S. C. Sharma and Menu Devi
1
1 1 (1 ) 0
( , ) (1 ) ;
p x y t t p
B x y =
∫
t − −t −e− − dt Re( ), Re( ), Re( )p x y >0 (3)Here, we extend Wright generalized Hypergeometric function using extended Euler’s
Beta function and the fact ( ) ( , )
( ) ( , )
k k
B k c c B c
γ γ γ
γ γ
+ −
=
− as follows:
1, 1,
( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; )
i i m j j n
a
m n b c z p
α γ
β
ψ
+ + 1
0 1
( )
( , )
1
( ) !
( )
m
k i i
p i
n k
j j j
a k
B k c z
c k
b k
α γ γ
γ β
∞ =
=
=
Γ + + −
=
Γ − Γ +
∏
∑
∏
(4) for Re( )p >0;Re( )c >Re( )γ >0.
Extended Riemann-Liouville fractional derivative operator: Ozarslan and Ozergin
defined the extended Riemann - Liouville fractional derivative operator as follows [10]:
2
, 1 ( )
0 1
{ ( )} ( )( )
( )
pz z
p t z t
z
Dµ f z f t z t µ e dt
µ
− − − −
= −
Γ −
∫
, Re( )µ <0, Re( )p >0, (5)and for m− <1 Re( )µ <m m( =1, 2,...).
,
{ ( )} { ( )}
m
p m
z m z
d
D f z D f z
dz
µ = µ −
2 1 ( ) 0
1
( )( )
( )
pz z
m
m t z t m
d
f t z t e dt
dz m
µ
µ
− − + − −
= −
Γ − +
∫
Forp=0, classical Riemann-Liouville fractional derivative operator can be obtained.
Lemma 1. If Re( ), Re( )v p >0then we have following integral representation [11]:
1 1
1 1 2 2
1 2 , 2 2 0
(1 ) ( ) ( )
p p
t
t t e dt p e W p
µ
µ ν
µ ν µ ν
− − −
− −
− −
− = Γ
∫
2. Main results
Theorem 1. Integral Representations of extended Wright generalized Hypergeometric
function is given as:
1, 1,
1, 1,
1
( , ) ,( ,1) 1 1 (1 ) ( , )
1 1 ( , ) ,( ,1) ( , )
0
1
( ; ) (1 )
( )
i i m i i m
j j n j j n
p
a c t t a
m n b c z p t t e m n b tz dt
c
α γ γ γ α
β β
ψ ψ
γ
−
− − − −
+ + =Γ −
∫
− (6)where Re( )p >0; Re( )c >Re( )γ >0;z a b, ,i j∈C; α βi, j R
+
∈ (i=1,..., ;m j=1,..., ).n Proof. Using equation (3) in (4), we get:
1, 1,
1
( , ) ,( ,1) 1 1 (1 ) 1
1 1 ( , ) ,( ,1)
0 0
1
( )
1
( ; ) (1 )
( ) !
( )
i i m j j n
m
p i i k
a k c t t i
m n b c n
k
j j j
a k
z
z p t t e dt
c k
b k
α γ γ γ
β
α ψ
γ β
∞ −
+ − − − − =
+ +
=
=
Γ +
= −
Γ −
Γ +
∏
∑ ∫
∏
(7)After interchanging the order of integration and summation and using equation (2), we obtain the desired result.
Corollary 2. If we take
1
u t
u
=
2
1, 1,
1, 1,
1 1 (1 )
( , ) ,( ,1) ( , )
1 1 ( , ) ,( ,1) ( , )
0
1 ( ; )
( ) (1 ) 1
i i m i i m
j j n j j n
p u
a u a
m n b c c m n b
u uz
z p e du
c u u
γ
α γ α
β β
ψ ψ
γ
+ − −
+ +
=
Γ −
∫
+ + (8)Corollary 3. If we take 2
sin
t= θ in Theorem 1, we obtain:
1, 1, ( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; )
i i m j j n
a
m n b c z p
α γ β ψ
+ +
2 2 1,
1, 1
( , )
2 1 2( ) 1 sin cos 2
( , ) 0
2
sin cos sin
( )
i i m j j n
p
a c
m n b
e z d
c
α
γ γ θ θ
β
θ θ ψ θ θ
γ
−
− − −
=
Γ −
∫
(9)For Wright generalized Hypergeometric function, we can easily obtain the following recurrence relations forl=1,...,n, as follows:
1, 1,
( , )
( , )
i i m m n
j j n
a
tz b
α ψ β
1, 1,
1, ; 1, ;
( , ) ( , )
( , ) ,( 1, ) ( , ) ,( 1, )
i i m i i m
l m n l m n
j j n j l l l j j n j l l l
a d a
b tz z tz
b b dz b b
α α
ψ β β β ψ β β
≠ ≠
= +
+ +
(10)
using equation (10) in equation (6), we obtain recurrence relations for extended Wright generalized Hypergeometric function as follows:
Corollary 4. For extended Wright generalized Hypergeometric function, we obtain the
following recurrence relations:
1, 1 1
1,
( , ) ,( ,1)
( , )
( , ) , ( ,1)
i i m m n
j j n a
z p b c
α γ
ψ β
+ +
1, 1 1
1, ;
( , ) ,( ,1)
( , )
( , ) ,( 1, ),( ,1)
i i m l m n
j j n j l l l
a
b z p
b b c
α γ
ψ β β
+ +
≠
= +
1, 1 1
1, ;
( , ) , ( ,1)
( , )
( , ) ,( 1, ), ( ,1)
i i m l m n
j j n j l l l
a d
z z p
b b c
dz
α γ
β +ψ + β β
≠
+
+
(11)
where Re( )p >0; Re( )c >Re( )γ >0; z a b, ,i j∈C; α βi, j R
+
∈ (i=1,..., ; ,m j l=1,..., ).n
Theorem 5. The Mellin Transform of extended Wright generalized Hypergeometric
function is given as:
1, 1 1
1,
( , ) , ( ,1)
( , ) ;
( , ) ,( ,1)
i i m m n
j j n
a
M z p s
b c
α γ
ψ β
+ +
1, 1 1
1,
( , ) , ( ,1)
( ) ( )
( , ) ,( 2 ,1)
( )
i i m m n
j j n
a s
s c s
z
b c s
c
α γ γ ψ
β γ + +
+
Γ Γ + −
= +
Γ − (12)
where Re( ), Re( )s p >0;Re( )c >Re( )γ >0; z a b, ,i j∈C;α βi, j R
+
∈
(i=1,..., ;m j=1,..., ).n
Proof: Taking the Mellin transform of extended Wright generalized Hypergeometric
S. C. Sharma and Menu Devi
1, 1 1
1,
( , ) ,( ,1)
( , ) ;
( , ) ,( ,1)
i i m m n
j j n
a
M z p s
b c
α γ
ψ β
+ +
1, 1, 1
( , )
1 1 1 (1 )
( , )
0 0
1
(1 )
( )
i i m j j n
p
a
s c t t
m n b
p t t e tz dt dp
c
α
γ γ
β
ψ γ
∞ −
− − − − −
= −
Γ −
∫
∫
(13)Interchanging the order of integration and using the fact that 1
0 ( )
s s ap a s p e dp
∞ − −
Γ =
∫
, weobtain:
1, 1 1
1,
( , ) ,( ,1)
( , ) ;
( , ) ,( ,1)
i i m m n
j j n
a
M z p s
b c
α γ
ψ β
+ +
1, 1, 1
( , )
1 1
( , ) 0
( )
(1 )
( )
i i m j j n
a s c s
m n b s
t t tz dt c
α
γ γ
β ψ γ + − + − −
Γ
= −
Γ −
∫
(14)using series representation of Wright generalized Hypergeometric function (2), after interchanging the order of integration and summation and using definition of Beta function, we obtain:
1, 1 1
1,
( , ) ,( ,1)
( , ) ;
( , ) ,( ,1)
i i m m n
j j n
a
M z p s
b c
α γ
ψ β
+ +
1 0
1
( ) ( )
( ) ( )
( ) !
( ) ( 2 )
m
i i k
i n k
j j j
a k s k
s c s z
c k
b k c s k
α γ
γ
γ β
∞ =
=
=
Γ + Γ + +
Γ Γ + − =
Γ − Γ + Γ + +
∏
∑
∏
(15)using definition of Wright Generalized Hypergeometric function, we obtain the required result.
Corollary 6. If we take s=1in equation (12), we obtain:
1, 1,
1 1 1 1
1, 1,
0
( , ) ,( ,1) ( , ) , ( 1,1)
( , ) ( )
( , ) , ( ,1) ( , ) , ( 2,1)
i i m i i m
m n m n
j j n j j n
a a
z p dp c z
b c b c
α γ α γ
ψ β γ ψ β
∞
+ + + +
+
= −
+
∫
(16)Corollary 7. If we take Inverse Mellin transform on both sides of equation (12), we
obtain:
1, 1 1
1,
( , ) , ( ,1)
( , )
( , ) , ( ,1)
i i m m n
j j n a
z p b c
α γ
ψ β
+ +
1, 1 1
1,
( , ) ,( ,1)
1
( ) ( )
( , ) , ( 2 ,1)
( )
i
i i m s
m n
j j n i
a s
s c s z p ds b c s
c
ν
ν
α γ
γ ψ β
λ + ∞
− + +
− ∞
+
= Γ Γ + −
+
Γ −
∫
(17)Theorem 8. For Re( )p >0; Re( )c >Re( )γ >0; z a b, ,i j∈C;α βi, j R
+
∈
(i=1,..., ;m j=1,..., )n , we have:
{
1,}
1,1, 1,
( , ) ( , ) ,( ,1)
, 1 1
( , ) 1 1 ( , ) ,( ,1) ( ; )
i i m i i m
j j n j j n
a a
p
z m n b m n b c
Dλ µ− zλ− ψ αβ z =zµ− +ψ + αβ γ z p .
Proof: From the definition of extended Riemann-Liouville fractional derivative (5), we
obtain:
{
}
21, 1,
1, 1,
( , ) ( , )
, 1 1 1 ( )
( , ) ( , )
0 1
( )
( )
i i m i i m
j j n j j n
pz z
a a
p t z t
z m n b m n b
Dλ µ zλ ψ αβ z tλ ψ αβ t z t λ µ e dt
µ λ
−
− − = − − − + − −
Γ −
∫
(18)using t=uzin (18) and equation (6), we obtain the required result:
Theorem 9. For extended Wright generalized Hypergeometric function, we obtain the
following derivative formula:
{
1,}
1,1, 1,
( , ) ,( ,1) ( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; ) 1 1 ( , ) ,( ,1) ( ; )
i i m i i i m
j j n j j j n
r
a a r r
m n b c m n b r c r
r
d
z p z p
dz
α γ α α γ
β β β
ψ ψ + +
+ + = + + + + , r∈N (19)
Proof. Using (2) in equation (6) and differentiating it with respect to z, we obtain
{
1,}
1, ( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; )
i i m j j n
a m n b c d
z p dz
α γ β ψ
+ +
1 1
1 1 (1 ) 1
1 0
1
( )
1
(1 )
( ) ( 1)!
( )
m
p i i k k
c t t i n k
j j j
a k
t z
t t e dt
c k
b k
γ γ α
γ β
− ∞
−
− − − − =
=
=
Γ +
= −
Γ − Γ + −
∏
∑
∫
∏
By taking k− =1 land using (6), we obtain:
{
1,}
1,1, 1,
( , ) ,( ,1) ( , ) ,( 1,1) 1 1 ( , ) ,( ,1) ( ; ) 1 1 ( , ) ,( 1,1) ( ; )
i i m i i i m
j j n j j j n
a a
m n b c m n b c
d
z p z p
dz
α γ α α γ
β β β
ψ ψ + +
+ + = + + + +
Continuing this process r times, we obtain the required result.
Theorem 10. For extended Wright generalized Hypergeometric function, we obtain the
following derivative formula:
{
1,}
1, ( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; )
i i m j j n
r
a m n b c r
d
z p dp
α γ β ψ
+ +
1, 1, ( , ) ,( ,1) 1 1 ( , ) ,( 2 ,1)
( 1)
( ; )
( 1)( 2)...( )
i i m j j n
r
a r
m n b c r z p
c c c r
α γ β ψ
γ γ γ
−
+ + −
−
=
− − − − − − (20)
Proof: Differentiating equation (6) with respect to p and using (2), we obtain:
{
1,}
1,1, 1,
( , ) ,( ,1) ( , ) ,( 1,1) 1 1 ( , ) ,( ,1) 1 1 ( , ) ,( 2,1)
( 1)
( ; ) ( ; )
( 1)
i i m i i m
j j n j j n
a a
m n b c m n b c
d
z p z p
dp c
α γ α γ
β β
ψ ψ
γ
−
+ + = − −− + + −
Continuing this process r times, we obtain the required result.
Theorem 11. For extended Wright generalized Hypergeometric function, we have:
1, 1, ( , ) ,( ,1) 2
1 1 ( , ) ,( ,1) ( ; ) i i m
j j n
a p
m n b c
S. C. Sharma and Menu Devi
1 , , 0
1
( ) ( ) ( )
1
( 1, 1)
( ) !
( )
m
r s i i k
i n r s k
j j j
L p L p a k
z
B r k s c
c k
b k
α
γ γ
γ β
∞
=
=
=
Γ +
= + + + + − +
Γ − Γ +
∏
∑
∏
(21)where Re( )p >0; Re( )c >Re( )γ >0;z a b, ,i j∈C;α βi, j∈R+ (i=1,..., ;m j=1,..., )n .
Proof: We will use the known identity [10]
2 1 1
(1 )
, 0
( ) ( ) (1 )
p
p r s
t t
r s r s
e e L p L p t t
− ∞
− + +
−
=
=
∑
− , 0< <t 1,using above result and series representation of Wright generalized Hypergeometric function (2) in (6) and after interchanging the order of integration and summation, we obtain:
1, 1, ( , ) ,( ,1) 1 1 ( , ) ,( ,1) ( ; )
i i m j j n
a
m n b c z p
α γ β
ψ
+ +
1 2
1 1 1 1 1
, , 0 0
1
( )
( ) ( ) (1 )
( ) !
( )
m
i i
p k
r k c s i
r s n r s k
j j j
a k
e z
L p L p t t dt
c k
b k
γ γ
α
γ β
− ∞
+ + + − + − + − =
=
=
Γ +
= −
Γ − Γ +
∏
∑
∫
∏
Multiplying both sides of above equation by e2 pand using definition of Euler Beta
function, we obtain required result.
Theorem 12. For extended Wright generalized Hypergeometric function, we have:
1, 1, 3
( , ) ,( ,1) 2
1 1 ( , ) ,( ,1) ( ; ) i i m
j j n
p
a m n b c
e +ψ + αβ γ z p
1
1 2
2 1 ,
, 0 2 2
1
( ) ( )
1
( )
( 1) !
( )
m
r k
r i i k
i
c r k r k n
r k
j j j
L p a k
z
p W p
c k
b k
γ
γ γ
α
γ β
+ + − ∞
=
− − − − + + =
=
Γ +
=
− − Γ +
∏
∑
∏
(22)Proof: Using (1 ) 1
p p p t t t t
e e e
− − −
− = − and the generating function of Laguerre polynomials, we
can easily obtain:
(1 )
0
(1 ) ( )
p p
p r
t t t
r r
e e e t L p t
− − ∞
− −
=
= −
∑
using above result and series representation of Wright generalized Hypergeometric function (2) in (6) and after interchanging the order of integration and summation, we obtain:
1, 1,
1
( , ) ,( ,1) 1 1
1 1 ( , ) ,( ,1)
, 0 0
1
( ) ( )
( ; ) (1 )
( ) !
( )
i i m j j n
m
p
r i i
p k
a i r k c t
m n b c n
r k
j j j
L p a k
e z
z p t t e dt
c k
b k
α γ γ γ
β
α ψ
γ β
−
− ∞
+ + − − =
+ +
=
=
Γ +
= −
Γ −
Γ +
∏
∑
∫
∏
Finally, using Lemma 1, we obtain the required result.
3. Special case
If we take m= =n 1,a1→c,α1→1,b1→βand β1→α in above theorems and corollaries,
we will obtain the results obtained by Ozarslan and Yilmaz [9].
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