Wright Type Hypergeometric Function and Its Properties
Snehal B. Rao1, Jyotindra C. Prajapati2, Ajay K. Shukla3 1Department of Applied Mathematics, The M.S. University of Baroda, Vadodara, India 2Department of Mathematical Sciences, Faculty of Applied Sciences, Charotar University of
Science and Technology, Anand, India
3Department of Applied Mathematics and Humenities, S.V. National Institute of Technology, Surat, India Email: [email protected], [email protected], [email protected]
Received January 7, 2013; revised February 6, 2013; accepted March 6,2013
Copyright © 2013 Snehal B. Rao et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
s v
Let s and z be complex variables, s be the Gamma function, and
ss
v
for any complex be the v
generalized Pochhammer symbol. Wright Type Hypergeometric Function is defined (Virchenko et al. [1]), as:
2 1
0
, ; ; ; k
k
a b
c
R a b c z
b c
!,k
k z
k k where
0,
; ,a b c, ; Re
a 0,Re
b 0, Re
c 0;which is a direct generalization of classical Gauss Hypergeometric Function 2 1F a b c z
, ; ;
2R a b c1 ;z
. The principal aim of this paper is to study the various properties of this Wright type hypergeometric function
, ; ; ; which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained; along with its relation-ship with Fox H-function and Wright hypergeometric function.Keywords: Euler Transform; Fox H-Function; Wright Type Hypergeometric Function; Laplace Transform; Mellin
Transform; Whittaker Transform; Wright Hypergeometric Function
1. Introduction and Preliminaries
The Gauss Hypergeometric Function is defined [2] as:
, ! 2,
k k k
k
a b
0
, ; ;
1, 0, 1,
k F a b c z
z c
zc k
; and (1.1)
The Generalized Hypergeometric Function, in a classi- cal sense has been defined [3] by
1 ; , , ;1
, ! 1 ;
p q
k p k
k 1
1
1 0 1 , , ;
, , , ,
1, p
p q p q
q
k
k q
k
a a z
F F a a
b b
a a
b b
p q z
b b z
z k
(1.2)
and no denominator parameter equal to zero or negative integer.
E. Wright [4] has further extended the generalization of the hypergeometric series in the following form
1 1
0 1 1
; ! p q
n
p p
n q q
z
n n z
n
n n
r
(1.3)
where and t are real positive numbers such that
1 1
1 q t p r 0.
t r
When r and t are equal to 1, Equation (1.3) dif-fers from the generalized hypergeometric function pFq by a constant multiplier only.
The generalized form of the hypergeometric function has been investigated by Dotsenko [5], Malovichko [6] and one of the special cases considered by Dotsenko [5] as
,
2R1 z 2 1R a b c, ; ; , ;z
0 !
n
n
a n b n
c z
a b n
c n
and its integral representation expressed as
(1.4)
11 1
0
1 c b 1 ad ,
b
c
t t zt t
(1.5) ,
2R1 z
c b b
where Re
Euler’s formul
Re 0
c b . This is the analogue of
a for the Gauss’s hypergeometric function 001 Virchenko et a
[3]. In 2 l. [1] defined the said Wright
Type Hypergeometric Function by taking 0
in
(1.4) as
2 1 2 1
0
, ; ; ;
!
k k
k
R z R a b c z
a b k
c
b c k k z ; 0,z 1.
(1.6)
If 1
, then (1.3) reduces to a Gauss’s hyper- geo-metric function. Galue et al. [7] and Virchenko et al. [1] investigated some properties of the function
2 1R a b c, ; ; ; z .
The following well-known facts have been prepared for studying various properties of the function
2 1R a b c, ; ; ; z .
Euler (Beta) transform (Sneddon [8]): of the function
The Euler transform f z is defined
z d Laplace transform (Sneddon [8]):
The Laplace transform of unction
as
: ,
1 a1
1
b1 .B f z a b
z z f z (1.7) 0the f f z
0
is de- fined as
e sz
d .L f z f z z
(1.8) Mellin transform (Sneddon [8]):The Mellin transform of the function f x
is de-fined as
; s1
d
,
0Re 0,
M f x s x f x x f
s(1.9) s
then
1
; 1
sd .2πiL
f x M f s x
f s x s (1.10) Wright generalized hypergeometric functio stava and Manocha [9]), denoted by
n (Sriva- pq, i
as
s defined
1, 1 , , p, p ;1 1 1 0
1
, , , , ;
! p q
q q
p k
i i i
q
k j j
j
A A z
B B
A k z
k B k
(1.11)
1 1 1,
, 1
1 1
1 , , , 1 ,
,
0,1 , 1 , , , 1 ,
p p p
p q
q q
A A
H z
B B
(1.12)
where
1 1 ,
,
1 1
, , , ,
, , , ,
p p m n
p q
q q
a a
H z
b b
denotes the Fox
ion [10].
2. Basic Properties of the Function
H-funct
2
R a b c
1,
z
Theorem 2.1
If , ,a b c; Re a 0, Re b 0, Re c 0; then
2 1 2 1
2 1
, ; ; ; , ; 1; ;
d
, ; 1; ;
d
c R a b c z c R a b c z
z R a b c z
z
(2.1 ).1
2 1 2 1
0
, ; ; ; , 1; ; ;
1 1
, ! 1 .
k k
k
R a b c z R a b c z
a b k
c z
a z
b c k k
b
(2.1.2)In particular,
2 1 2 1
1
, ; 1; , 1; 1;
1, ; ;
c
F a b c z F a b c z
az
F a b c z
(2.1.3)
Proof. 2 1
2 1
1 0
0
0
2 1 2 1
d
, ; 1; ;
d 1
1 !
! 1
1 !
, ; ; ; , ; 1; ; ,
k k
k
k k
k
k k
k
z R a b c z
z
a b k
c k z
z
b c k k
a b k
c c z
b c k k
a b k
c c z
b c k k
c R a b c z c R a b c z
which is the (2.1.1). Now,
2 1 2 1
0
0
0
1
, ; ; ; , 1; ; ;
! 1
1 !
1 b
1 1
1
! 1
1
1 1 1 !
1
k k
k
k k
k
k k
k
k k
k
R a b c z R a b c z
a b k
c z
b c k k
a b k
c z
b c k k
c
a b k z b k
c k k b
a b k
c z
b b c k k
a z b
c
1 1
1
0
0
1 1 1
1 1 1 !
1 1
1 1
!
1 1
, !
k k
k
k k
k
k k
k
a b k z
b c k k
c a z
b b
a b k z
c k k
a b k
c z
a z
b c k k
This is the proof of (2.1.2).
For c1 and substituting 1 in above result, this
will immediatel s to particul .1.3).
Theorem 2.2 1)If
0, Re 0
c
ar case (2 y lead
, , , ;
Re 0, Re 0, Re
and then
a b c
a b
1
1 1
c
u
2 1 0
1 , d
, ; ; ;
u R a zu u
a b c z
(2.2.1)
2 1
c R
; ; ; c
b c
2)If
, , , , ;
Re 0, Re 0, Re 0, Re 0
and then
a b c
a b c
1 1
2 1 1
2 1
, ; ; ; d
, ; ; ;
x
c
t c c
x
.
s s t R a b c s t s
x t c R a b c x t
(2.2.2) 3)
, , ;
Re 0, Re 0, Re 0
and then
a b c
a b c
1 2 1 0
2 1
, ; ; ; d
, ; 1; ; .
z c
c
t R a b c t t
z
R a b c z
c
(2.2.3)
In particular,
1 2 1 0
2 1
, ; ; d
, ; 1; .
z c
c
t F a b c t t
z
F a b c z
c
2.2.4)
Proof. 1)
(
1
1 1
2 1 0
1
1 1
0 0
1
1 1
0 0
0
1 , ; ; ;
1 d
1 d
!
!
c
k
c k
k
k c k k
k
k k
k
c
u u R a b c zu
a b
c
u u
b c
a b k c z
u u u
b c k k
a b k
c z
c
b c k k
c 2 1R a b c
, ; ; ; .z
which concludes the proof of (2.2.1).
du
! zu k
u
k k
c
c
2)
1 1
2 1 1
1
1 2 1 1
1 1 1
2 1 0
, ; ; ; d
, ; ; ;
1 , ; ; ;
applying the transformation formula c
t
x
c
t
c c
c
x s s t R a b c s t s
x t
c x t s t
s t R a b c s t
x t
c
u u x t R a b c u x t
s t u
x t
c b
d
d , x
s
x t u
1
1 1
0 0
1
1 1
0 0
0 2 1
1 d
!
1 d
!
!
, ; ; ;
k k k
c
c k
k
k k
c k c k
k
k k
c k
k
c
a b k
c u x t
u u x t u
c k k
a b k
c c x t
x t u u u
b c k k
a b k
c x t
x t c
b c k k
x t c R a b c x t
.Therefore,
1
12 1
, ; ; ;
d
1
2 1 , ;c c
t c
x s s t R a b c s t s x t c R a b c
Which is the proof of (2.2.2).
; ;
.x
x t
3)
1 1
2 1
0 0
0 0
2 1 0
, ; ; ; d d
! !
1
, ; 1; ; .
1 !
k
z z k
c c k k
k k
k
c c
k
k
t
a b k a b k
c c
t R a b c t t t t
b c k k b c k
z
a b k
c
z z
R a b c z
c b c k k c
This leads the proof of (2.2.3).
1 0
d z
c k
t t
k
On putting 1, in the above expression immediately leads to (2.2.4).
Theorem 2.3
c 0, then, , ;Re 0, Re 0, Re
a b c a b
If
, ; ; ;
, ; ; ;
d
c c m c
R a b c z z R a b c m z
z c m
1 1
2 1 2 1
d m
z
(2.3.1)
Proof.
1 1
2 1
0
1
1 1
0
, ; ; ;
d d !
1 2
!
!
m m k k c
c k
k
k k c m
k
c m k c m
k
a b k
c z
z R a b c z
z z b c k k
a b k
c k c k c k c m z
k
z
a b k
c c m c
z z
c m b c m k k
d d
0
k k
b c k
2 1R a b c, ; m; ; z .
c m
This establishes (2.3.1).
3. Representation of Wright Type
Hypergeometric Function
,
2 1R a b c, ; ; ; z
2
R a b c
1
z
in Terms of the Functio
U
n
pF
qsing the definition
2
0 !
k
b c k k
1 , ; ; ;
k k
a b k
c z
R a b c z
, and taking
q
we have
2 1
1
0 0
1
1
1
, ; ; ;
1 1
, , , , ;
1 1
, , , ;
, ; ;
, ; ;
q
k i k qk
q
k qk k
j
q q
q q
R a b c q z
1
! 1 !
qk
k k
k
qk
k
b i
a q
b z q z
c k c j k
q q
b b b q
a
q q q
F z
c c c q
q q q
a q b
F z
q c
(3.1)
a
where q b; is aq-tuple b b 1 b q 1
q
;
, , ,
q q
q c; is a q-tuple c c, 1, ,c q 1
q
. Convergence criteria for genera
ction
q q
lized hyperfeometric fun
1 1, 2, ,
F 0
1 2 1
; , , , ;
k p
p k k
p q
k
q k
z
z
:
1) If pq function
! q k k
, the pFq converges for all finite .
f p q 1 function
z
2) I , the pFq converges for
1
z and diverges for 1.
If function
z
3) p q 1, the pFq is divergent for
0 .
If p q 1 nction
z
4) , the fu pFq is absolutely
con-n circle
verge t on the 1 if
1 1
0
q p
j i
j i
.4. Mellin-Barnes Integral Representation of
,
zRe
2
R a b c
1
z
Let
;Re a 0,
Then
Theorem 4.1
is represented by the Mellin- Barnes integral
2 1 , ; ; ;
1
d 2πi
s
L R a b c z
c s a s b s
z s
b a c s
(4.1.1)
0, ; , ,
Re 0, Re 0.
a b c
b c
where arg
z π; the contour of integration beginning at i and ending at i , and intended to separate the poles of the integrand at s k k, 0,1,to the left and all the poles at s n a n, 0,1,as well as, 0,1,
n b
s n
to the right.
Proof. We shall use the sum of residues at the poles
, 0,1,
sk k to obtain the integral of (4.1.1).
2 1
0
0 , , , ;
!
! 1
1 k k
k
k
k
k k
k R a b c z
a b k
c z
b c k k
c a k b k z
b a c k k
c b k
a k z
c k
(4.1.2) Now
0 1
b a k
,
0
2πi
π
sinπ
1 1
L
s k
k
k
a c s
c s a s b s z
res
b a c s
c
s
b k
a k z
b a k c
0
k
0 1
1
k
k k
s s c s
c
1
d
s
c s a s b s z
s b
1
lim s
s
k b s
a s z
s
b a
k
(4.1.3)
5. Integral Transforms of
2 1
,
(4.1.2) and (4.1.3) completes the proof of (4.1.1).
R a b c
z
discussed some useful integral trans- forms like Euler transforms, Laplace transfor , Mellin transform and Whittaker transform.Theorem 5.1 (Euler (Beta) transforms). In this section we
m
1 2 1 3 2,1 , , , , ;
,
, , , ;
R a b c
a b x
c c a b (5.1.1) 1 1 0
1 , ; ; ; d
z z xz z
where , , , , , , ;Re 0, Re
Re 0, Re 0, Re 0, Re
a b c a
c
0, 0, Re b
0. Proof.
1 1 1 2 1 01 , ;
z z R a b
1 0 1 1 1 0 0 0 3 2
; ; d
1 d
!
!
,1 , , , , ;
k k k k k k k
c xz z
xz
b k
a c b k x
z z z
c k b k
c a k b k k
1 1 0 1 d ! k k
z z z
c k b k
a c
x
a b c k k k
a b c a b
, , ,
; , x c
This is the pr f (5.1.1).
Rem 1
oof o Puttingark: in (5.1.1), we get
1 1 1 2 1
3 21 , ; ; d
,1 , ,1 , , ;
.
,1 , , ;
z F a b c xz z
a b x
c c a b (5.1.2) 0
z
Taking , c and substituting in place of the notation ; (5.1.1) re
; ; ; d
; ; , xz z x (5.1.3 duces to
1 1 1 2 1 0 2 1 1 , , , cz z R a b c
c R a b c
Also, considering ) and c in (5.1.1), with
replacement of z by
1
1 c
z at 2R1, we get
; ; ; 1 d
; , .
c x z z
x (5.1.4)
2 1 0 2 1 1 , , , ;z z R a b
c R a b c
1
Theorem 5.2 (Laplace transform).
, ;
1 2 1
e sz , ; ; ; d
z R a b c xz z
0
3 1
,1 , , , , ;
, x a b s c s a b (5.2.1) c
where , , , , , ; Re 0, Re 0,
Re 0, Re 0, Re 0, Re 0,
Re 0 and 1.
a b c a b
c s x s Proof.
1 2 1 0 1 0 0 1 0 0 0 3 1e , ; ; ; d
e d
!
e d
!
!
,1 , , , ,
sz k sz k k k sz k k k k
z R a b c xz z
xz
a c b k
z z
c k b k
c a k b k x
z z
a b c k k
c s a k b k k x
a b c k k s
a b s c a b
; , , ; x s c This is the proof of (5.2.1).
Theorem 5.3 (Mellin transform). 1
2 1 0
, ; ; ; d
, s
s
t R a b c t t
c s a s b s
a b c s
(5.3.1)
here , , 0, Re 0,
Re 0, Re 0, Re 0.
a b a b
c
sw c, , s ; Re
Proof. Putting zt in (4.1.1), we get
2 1 , ; ; ;
π
1
d 2πi
1
s s
L
s
R a b c t
s b
c s a s b s
t s
b a c s
(5.3.2)where,
1 d
2 i
s
L
c a s s
t s
b a c s
d , 2πiLf s t s
s
.c s a s b s
f s
a b c s
Theorem 5.4 (Whittaker transform).
1 1 2
, 2 1
0
4 2
e , ; ; ; d
1
,1 , , , , ;
2 ,
, , 1 ,
pt
t W pt R a b c t t
a b
p c
p b
c
(5.4.1)
0, Re 0,
0. b
Proof. To obtain Whittaker transform, we use the fol-lowing integral:
a
where , , , , , , ; Re
Re 0, Re 0, Re 0, Re
a b c p a
c
1 2
, 0
e d
1
t v
t W t
1 1
2 2 ,
t
v v
v
where 1.
2 v Re
Substituting ptv on the L. ces to
H.S. of (5.4.1), it re-du
1 1
2
, 2 1
0
0 1
1 2
, 0
1
e , ; ; ; d
1
e d
!
1 1
2 2
1
v
k
v k
v v
W v R a b c v
p p p
c p a k b k
a b c k p
v w v v
k
c p a k b k
c
k k
0 k
a b k
k
4 2
1 !
1
,1 , , , ;
2 .
, 1 , ;
k
k k p
a b
c p
p
a b
c
This completes the proof of (5.4.1).
6. Relationship with Some Known Spec
Functions (Fox H-Function, Wright
ergeometric Function)
6.
.1 ), we get
ial
Hyp
1. Relationship with Fox H-Function Using (4 .1
2 1R a b c, ; ; ; z
1,22,2
1d 2πi
1 ,1 , 1 ,
. 0,1 , 1 ,
s
L
c s a s b s
z s
b a c s
a b
c
H z
c
b a
6.2. Relationship with Wright Hypergeome Function
The Generalized Hypergeometric Function
tric
2 1R a b c, ; ; ; z as in (1.3) is
2 1R z 2R a b c, ; ; ; z
1
0
; 0, 1.
!
k k
k
a b k
c
z z
b c k k
(6.2.1)
From (1.11) and (6.2.1) yields
2 1 2 1
2 1
, ; ; ; ,1 , , ;
, ;
R z R a b c z
a b z
c
c
a b
.
(6.2.2)
dgem
The authors are thankful to the reviewers for their valu- able suggestions to improve the quality of paper.
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