Vol. 09, Issue 1 (January. 2019), ||V (II) || PP 55-60
On Some Double Integral Transform of Aleph
( )
-Function
Yashwant Singh
Department of MathematicsGovernment College, Kaladera, Jaipur (Rajasthan), India Corresponding Author: Yashwant Singh
Abstract:
In the present paper , the author will establish a double integral transform of Aleph
-function which leads to yet another interesting process of augmenting the parameters in the
-function. The result is of general character and on specializing the parameters suitably yields several interesting results as particular cases.Keywords:
-function, Euler Transformation,H
-function, Hypergeometric Function, IntegralTransformation.
(2010 Mathematics subject classification: 33C99)
--- --- Date of Submission: 04-01-2019 Date of acceptance: 19-01-2019 ---
---I.
INTRODUCTION
Rainville [6, p.104], Abdul Halim and Al-Salam [1] have shown that the single and double Euler transformations of the hypergeometric function p
F
qare effective tools for augmenting its parameters. Srivastava and Singhal [9] and Srivastava and Joshi [10] have discussed some similar interesting properties ofp
F
qin doubleH
-function and double Whittaker transforms respectively.In what follows for the sake of brevity, we have used the symbols
( ,
a
r
r), ( , ), ( ,
r a
r
a
), (( ,
r a
p))
todenote the set of parameters
( ,
a
1 1),..., ( ,
a
r r); ,
a a
1
,...,
a
r
1
; ( , ), ( ,
r a
r
a
)
r
r
r
and1 2
( ,
r a
), ( ,
r a
),..., ( ,
r a
p)
respectively.The
- function introduced by Suland et.al. [11] defined and represented in the following form:
,, : :
[ ]
i i i m n p q r
z
z
, 1, 1,, ; :
1, 1,
(
,
) ,[ (
,
)]
( ,
)
,[ (
,
)]
i i i
i j j n i ji ji n p m n
p qi r
j j m i ji ji m q
a
a
z
b
b
1
( )
2
s
L
s z ds
(1.1)
Where
1
;1 1
1 1 1
(
)
(1
)
( )
(1
)
(
i i
m n
j j j j
j j
q p
r
i ji ji ji ji
i j m j n
b
s
a
s
s
b
s
a
s
(1.2)
We shall use the following notation:
*
1, 1,
,
,[
,
]
i j j n i ji ji n p
A
a
a
, *
1,1,
,
,[
,
]
i j j m i ji ji m q
B
b
b
*
1, 1,
,
,[
,
]
i j j f i ji ji f u
C
a
a
, *
1,1,
,
,[
,
]
i j j g i ji ji g v
II.
MAIN RESULT
In this section, we have established the following double integral transform of
-function: Ifs k
,
andr
are positive integers, then1 1 , * , *
, : : * , : : *
0 0
(
)
(
)
(
)
i i i i i i
f g C m n s k r A
u v r D p q r B
x
y
x
y
x
y
tx y x
y
dxdy
=1 1
1 1
1
1 1 1 ( ) 2 2
(1 )
2 2 2 2
1 2
(2 )
(
)
v u
j j
d c A u v
D f g u v
s
k
D
s
k
1 1
( )
(( ,1 )), ( ,1 ), ( ,1 ), *, ( ,1 ),..., ( ,1 ) ,
, : : (( ,1 )), *, ( ,1 ), ( ,1 ),..., ( ,1 )
,
f f u
i i i i i g g v
D v u
D A D s k A D A d D A d
m Dg n Df
p Dv q Du r D D A C B k s D A c D A c
t D
(2.1)Where
)
,
,
,
,
(
)
s k
s k
s k
s
k D
s
k
r A
s
k
0
Dg
Du
Dv
Du
q
p u
,
v
2
g
2
f
2 , 0
v
n
p p
,
q
2
n
2
m
2 ,
q
Re min
i jimin
i jiRe(
)
Re
,
,
l t1
ji ji
d
b
s
k
D
A
D
a
C
D
s
k
1, 2,..., ;
1, 2,..., ;
1, 2,... ;
1, 2,..., ; , Re(min
i)
,
i
f j
m l
n t
g u
C
A
v
1
Re max
1
(
), Re max
,
,
,
2
j
l j
d
s
k
A
uD v
D Dv
Du
D Dv
Du
a
s
k
1
1
1, 2,..., ;
1, 2,..., ;
1, 2,..., ;| arg
|
,
2
2
i
u j
v l
u
f
g
u
v
1
1
| arg |
, Re
0, Re
0,
1, 2,...,
2
2
ji ji
i i i
ji ji
b
b
t
m n
p
q
s
k
j
m
And the double integral converges.
Proof: To prove (2.1), we start with the following known result [2, p. 177]
1 1 1
0 0 0
(
x
y x
)
y
dxdy
B
( , )
( )
z z
dz
(2.2) Which is valid forRe( )
0
andRe( )
0
.It is easy to prove by following the technique of reversing the order of integrations, that
1 1
2 2
1 1 , *
, : : * 1
0 0 2
(
)
(
)
2
(
)
i i i
m n s k r A
p q r B
s
k
x
y x
y
tx y x
y
dxdy
s
k
1 , ( ,1 ), ( ,1 ), *
, : : *, ( ,1 )
0
( )
i i i
m n D s k A
p q r B k s
z z
t z
dz
(2.3) Wheres k
,
andr
are positive integers,1
1
,
,
,
2(
),| arg |
,
(
)
2
2
s k
i i
s k
s k
s
k D
s
k
r p
q
m n
t
m n
p
q
s
k
Re
i ji0, Re
i ji0,
1, 2,..., .
ji ji
b
b
s
k
j
m
, *
, : : *
( )
i i i
f g C
u v r D
z
z
z
And evaluating the integral on the right hand side using [8, p.401] the result (2.1) follows.
III.
PARTICULAR CASES
On choosing the parameters suitably in (2.1), several known and unknown results are obtained as particular cases. However, we mention some of the interesting results here.
(a) Taking
1 1 2
1
1
2,
0,
1,
,
,
,
,
1,
1,
1
2
2
i j j j j i
f
v
g
u
c
d
v d
v
r
in (2.1) and using [3, p.216, (5)]1,1 1 1
2,0 2 2 2
1,2 ( ,1),( ,1)
1
,
2
x
b b b
H
x
e
K
x
We obtain
1 1 ( )
1 1 2 2 , *
, *
0 0
1
(
)
(
)
(
)
2
x y m n s k r A
v p q B
x
y
x
y
a
K
x
y
H
tx y x
y
dxdy
=1 1
1 1 1 1 2 2
(2 )
1
2 2 2 2
1 2
(2 )
(
)
D f g u v
A
s
k
D
s
k
, 2 (( ,1 )), ( ,1 ), ( ,1 ), *
2 , 1
*, (( , )), ( ,1 ) 2
,
D
m n D D A v s k A
p D q D D
B D A k s
t D
H
(3.1)
Where
,
D
and
have the same value as (2.1) and1
;
2(
), Re(
)
0, Re(
)
0,
2
j j
j j
b
b
A
p
q
m n
s
v
s
v
1
1
1
Re
0,
1, 2,..., ; Re( )
0,| arg |
.
2
2
2
j
j
b
v
D
j
m
t
m n
p
q
(b) Further, replacing
q t
,
and
a
p,
p
byq
1,
t
and
1
a
p,
p
respectively and then putting1 1
1,
,
0,
j1
j(
1, 2,... )
m
n
p b
b
b j
q
, using the result [3,p. 215, (1)] and [3,p. 4, (11)], we obtain an interesting result obtained by Srivastava and Singhal [9]:
1 1
( ) ,
1 1 2 2
, 0 0
1
(
)
(
)
:
(
)
2
p p q q
x y a
s k r
v p q b
x
y
x
y
a
K
x
y
F
tx y x
y
dxdy
=
1 2
1
2
( , )
1
v
B
1
( , ,), ( , ), ( , ), ,1 2
3 3 2 2 2 ,1 , ( , ), ( , 1)
p
q s k r
s k r v s k a
p s k r q s k r b s k k s
s k
r
F
t
, (3.2)provided
Re(
1
)
0, Re( )
0, Re( )
0.
2
v
(c) Setting
2,
0,
1,
11
,
11
,
21
,
1,
1
2
2
i1 ,1 1
2,0 2
1,2 1 1 ,
( ,1),( ,1)
2 2
,
x k
k m
m m
H
x
e
W
x
We have
1 ( )
1 1 2 , *
, , *
0 0
(
)
x y v[ (
)]
p qm n s k(
)
r BAx
y
x
y
e
W
x
y H
tx y x
y
dxdy
=1 1
1 1 2 2
(2 )
2 2
1 2
(2 )
(
)
D A
s
k
D
s
k
1
( , ), ( ,1 ), ( ,1 ), *
, 2 2
2 , *, ( ,1 ), ( , )
,
D
D A v s k A
m n D
p D q D D B k s D A
t D
H
(3.3) Where, ,
D
andA
are given in (2.1);1
1
2(
),| arg |
, Re( )
0, Re(
0,
2
2
jp
q
m n
t
m n
p
q
sb
1
Re(
)
0, Re
0,
1, 2,..., .
2
j j
k
sb
m n
Db
v
j
m
(d) Further, replacing
q t
,
and
a
p,
p
byq
1,
t
and(1
a
p,
p)
respectively and then putting1 1
1,
,
0,
j1
j(
1, 2,..., )
m
n
p b
b
b j
q
and using the result [p.215,(1)], (3.3) reduces to a result due to Srivastava and Joshi [10,p.19,(2.3)]
1
( ) ,
1 1 2
, ,
0 0
(
)
(
)
p p:
(
)
q q
x y a s k r
v p q b
x
y
x
y
e
W
x
y
F
tx y x
y
dxdy
=
1
2
( , )
1
v
B
1
( , ,), ( , ), ( , ), ,1
' 2
3 3 2 2 2 ,1 , ( , ), ( , 1)
p
q
s k r v s k a
p s k r
F
q s k rt
b s k k s r
(3.4)Where
'
,
(
)
s k r s k
s k
s k
s
k
r
s
k
1
Re( )
0, Re( )
0, Re( )
0, Re
0
2
v
and the resulting hypergeometric series converges.With
0,
1
2
v
and1
2
,(3.4) reduces to the earlier results of Jain [5] and Singh [7].(e) Choosing
1 1 2
1
1
1,
2,
1
,
,
,
1,
1,
1
2
2
j j j j i1 ,1 1
1,1 2
1,2 1 1 ,
( ,1),( ,1)
2 2
1
2
,
(2
1)
x k
k m
m m
k
m
H
x
e
M
x
m
We obtain
1 ( )
1 1 2 , *
, , *
0 0
(
)
x y k m[ (
)]
p qm n s k(
)
r ABx
y
x
y
e
M
x
y H
tx y x
y
dxdy
=1 1
1 1 2 2
(2 )
2 2
1 2
(2
1)
(2 )
1
(
)
2
D k A
m
s
k
D
k
m
s
k
1 1
( , ), ( ,1 ), ( ,1 ), ( , )
, 2 2
2 , ( ,1 ), ( , )
,
D
D A m s k D A m
m D n D
p D q D D k s D k A
t D
H
(3.5) Where, ,
D
andA
are given in (2.1);1
1
2(
),| arg |
, Re( )
0, Re(
)
0,
2
2
jp
q
m n
t
m n
p
q
sb
1
Re(
)
0, Re
0,
1, 2,..., .
2
j j
kb
Db
m
j
m
(f) Substituting
1,
0,
2,
11
,
21
,
1,
1
2
2
if
g
u
v
d
v d
v
r
and using the result [3,p.216,(3)]1,0
0,2 1 1
,1 , ,1
2 2
(2
)
v
v v
H
x
J
x
,
(2.1) reduces to
1 1 , *
, *
0 0
(
)
(2
(
)
m n s k Av p q B
x
y
x
y J
x
y H
tx y
dxdy
=
1 1
2 1 2 2
1 2
2
(
)
A
D
s
k
s
k
1 1
,1 , ( ,1 ), ( ,1 ), *, ,1
, 2 2
2 , *, ( ,1 )
D
D A v s k A D A v
m n D
p D q p B k s
D
H
t
(3.6)
Where
, ,
D
andA
have the same values given in (2.1);1
1
2(
),| arg |
, Re( )
0, Re(
)
0,
2
2
jp
q
m n
t
m n
p
q
sb
1
Re(
)
0, Re
0,
1, 2,..., ;
2
j j
kb
v
Db
j
m
1
Re
, ,
1, 2,..., .
4
i
D
Da
i
n
International organization of Scientific Research
60 | Page
REFERENCES
[1]. Abdul-Halim,N. and Al-Salam, W.A.; Double Euler transformation of certain hypergeometric functions, Duke Math. J., 30(1963), 51-62.
[2]. Edwards, J.; A Treatise on Integral Calculus, Chelsea, New York (1954).
[3]. Erdelyi, A. et. al. ; Higher Transcendental Functions, vol. I, McGraw-Hill, New York (1953).
[4]. Fox, C.; The G and H function as symmetric Fourier kernels Trans. Amer. Math. Soc. 98 , 1961 , 395-429.
[5]. Jain, R.N.; Some double integral transformations of certain hypergeometric functions, Math. Japon, 10(1965), 17-26.
[6]. Rainville, E.D.; Special Functions, Macmillan, New York (1960).
[7]. Singh, R.P.; A note on double transformation of certain hypergeometric functions, Proc. Edinburgh. Math. Soc. (2),1491965), 221-227.
[8]. Saxena, R.K.; Some theorems on generalized Laplace transforms, Proc. Nat. Inst. Sc. India, 26 A(1960), 400-413.
[9]. Srivastava, H.M. and Singhal, J.P.; Double Meijer’s transformation of certain hypergeometric functions, Proc. Camb. Phil. Soc. 64(1968).
[10]. Srivastava, H.M. and Joshi, C.M.; Certain Double Whittaker transforms of generalized hypergeometric functions, The Yokohama Mathematical Journal, vol. XV, (1) (1967), 17-32.
[11]. Sudland, N., Baumann, B. and Nonnenmacher, T.F.; Open problem: who knows about the Aleph(
