International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 1 ISSN 2250-3153
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Certain Case of Reducible Hypergeometric Functions of
Hyperbolic Function as Argument
Pooja Singh*, Prof. (Dr.) Harish Singh
**
* Research Scholar, Department of Mathematics, NIMS University, Shobha Nagar, Jaipur (Rajasthan)
** Department of Business Administration, Maharaja Surajmal Institute, Affiliated to Guru Govind Singh Indraprastha University, New Delhi
Abstract- In this paper, we specialized parameters and argument, Hypergeometric function FE (1, 1, 1, 1, 2, 2; γ1, γ2, γ3; cosh x,
cosh y, cosh z) FG, Fk and FN can be reduced to the hypergeometric function of Bailey’s F4(1, 2, γ2, γ3; -coshy, - cosh z) and also
discussed their reducible cases into Horn’s function. In the journal we consider hypergeometric function of three variables and obtain
its interesting reducible case into Bailey’s F4 & Horn’s function.
In the section 2, hypergeometric function of four variables can be reduced to the hypergeometric function of one, two & three
variables with some new and interesting particular cases.
Index Terms- Matrix argument, Confluent, Hypergeometric function, Beta and Gamma integrals, M-transform
I. ON HYPERGEMOTERIC INTEGRALS
1.1 INTRODUCTION
e will study Laplace’s double integral for Saran’s function FE (1, 1, 1, 1, 2, 2; γ1, γ2, γ3; cosh x, cosh y, coshz) which has
been reduced to Bailey’s F4 (1, 2, γ2, γ3; - cosh y, - cosh z) and pochhamer type of Integrals for FE, FG, FK and FN and also
discussed their reducible cases into Horn’s functions. The purpose of studing only the function FE, FG, FK and FN is mainly due to the
function in their integral representation contain Appell’s function F1 or F2 or the product of Gauss’s hypergeometric series which can
be reduced by the following relations.
2F1 (, ; γ; cosh y) = m !(γ
(β
n
nn
k,m )
))(
0
cosh m y (1.1.1)
2F1 (, ; γ; cosh y) = mm !(γ
ee(β
n
myy
nn
k,m 2 )
)())(
0
(1.1.2)
2F1 (, ; γ; cosh y) =
1
0
11 )cosh1()1()()
)(dtyttt
(γ
(1.1.3)
2F1 (, ; γ; cosh y) = (1- cosh y)-
2F1 (,γ-; γ; 1cosh
cosh
y
y
) (1.1.4)
Similarly
2F1 (, ; γ; cosh y) = (1- cosh y)-
2F1 (γ - , ; γ; 1cosh
cosh
y
y
) (1.1.5)
2F1 (, ; γ; cosh y) = (1- cosh y)γ--
2F1 (γ-, γ-; γ; coshy y) (1.1.6)
2F1 (, ; γ; cosh y) = (1- cosh y)-
(1.1.7)
F1(,,’,: cosh x, cosh y) = (1 – cosh y) -
(1 – cosh y) -
(1.1.8)
W
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 2 ISSN 2250-3153
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F1(,,’,,’; cosh x, cosh y) = (1 – cosh x – cosh y) -
(1.1.9)
F2(,,’,,’; cosh x, cosh y) = (1–cosh y) - -
(1–cosh x–cosh y)-
(1.10)
where
|cosh x| <1 | cosh y| <1|Cosh z|<1 for (1.1.1) to (1.1.10)
1.2 Reduction of integrals of FE(.) INTO BAIEY’S F4 (.)
The hypergeometric function FE is defined by
F1 (1, 1, 1, 1, 2, 2; γ1, γ2, γ3; cosh x, cosh y, coshz)
= !!!)()()(
coshcoshcosh)()()(
321
211
0,, pnm
zyxb
pnm
pnm
pnmpnm
pnm
(1.2.1)
absolutely convergent if m+,1)( 2 pn|coshx|<1, |coshy|<1,|coshz|<1 and |coshz|<1 and It’s integral representation S. Saran
(1957) is given by
FE =
0
111111
021
)cosh;;()()
12
1
xpFtpe(
tp
X0F1 (γ2;- pt cosh y)0F1(γ3;- pt cosh z) dpdt (1.2.2)
Where R (1)> 0 and R (2) > 0
F4 (1, 2, γ2, γ3; - cosh y, - cosh z)
=
0
21011
021
)cosh;()()
12
1
xpFtpe(
tp
X0F1 (γ3;- pt cosh y) dp dt (1.2.3)
Changing the variable t to T by the substitution t = C2T
2/4p and writing
1 = ½ (++-) 2 = ½ (+++)
γ2 = +1 γ3 = +1, cosh y = 2
2
c
a
, cosh z = 2
2
c
b
(1.2.4)
F4
2
2
2
2
,1,1),(2
1),(
2
1
c
b
c
avvv
=
)(2
1)(
2
12 )1(
)(
vv
e
v
v
0
22
10
22
10)1(1P4
0
)4
1;1()
4
1;1(
22
TbvFTaFTpe v
TCp
dpdt
We know that [E.T. Wittaker and G.N. Watson (1902)]
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 3 ISSN 2250-3153
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0 exp{-(cx + a / x)}dx = 2(c / a)
v2
1
a) / (cKv(2
ac ), (1.2.5)
so the p- integral
0
1P4 )2()2
(2
22
ackCT
dppe p
TCp
(1.2.6)
and changing 0F1 in to Bessel function of the kind the relation
Jk(z)= )1(
)2
1(
k
z k
0F1
)4
1;1( 2zk
(1.2.7)
0
1
1
)1()1(
(2
1(
2
12
)()()(vc
vvba
dtctJbtJatJtv
v
v
F4
2
2
2
2
,;1,1),(2
1),(
2
1
c
b
c
avpvpv
(1.2.8)
Where F4 is fourth type of Appell’s function
1.3 REDUCTION OF FE, FG, FK AND FN INTO HORN’S FUNCTION
Let us consider the integral S. Saran (1955) for FE viz.
FE=
c
tt 32 )1()()i2(
)2()()(2
3232
F2 (1,1, 2,γ1, γ2 + γ3-1; cosh x,
)1
coshcosh
t
z
t
y
dt (1.3.1)
where
|cosh x|+t
z
t
y
1
coshcosh
< 1 along the contour.
Puttting 1 = γ1 and 2 = γ2 + γ3-1 in equation (1.3.1) we get
FE=2
3232
)i2(
)2()()(
(-t)-γ2
(-t) –γ3
(1-cosh x - t
z
t
y
1
coshcosh
)-1
dt
We can expand
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 4 ISSN 2250-3153
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(1-cosh x - t
z
t
y
1
coshcosh
)-1
= (1-cosh x – cosh y) -1
mm
nm
onm tyx
z
t
t
yx
y
nm
1
1.
coshcosh1
cosh1.
coshcosh1
cosh
!!
)( 1
,
….. (1.3.2)
Where
t
y t1.
ycosh cosh x1
cosh
<1,
t
z
1
1.
ycosh cosh x1
cosh
< 1
along the contour
= (1 – cosh x – cosh y – cosh z) -1
mm
nm
onm tyyx
z
t
t
zyx
y
nm
1
1.
coshcoshcosh1
cosh1.
coshcoshcosh1
cosh
!!
)( 1
,
… (1.3.3)
where
t
y t1.
zcosh -ycosh cosh x1
cosh
< l and,
t
z
1
1.
zcosh -ycosh cosh x1
cosh
< 1
along the contour using (1.3.2) and (1.3.3) and then evaluating the integral, after changing the order of the integration and summation,
keeping
)()1()1(
)2()1()(
211
idttt
c
We will get,
FE(1, 1, 1, γ1, γ2 + γ3 -1, γ2 + γ3 -1, γ1, γ2, γ3, : cosh x, cosh y, cosh z)
= (1 – cosh x – coxh y) -1
.
H1(1- γ3, 1, γ2 + γ3-1, γ2; 1-ycosh cosh x
cosh,.
1-ycosh cosh x
cosh
zy
) (1.3.4)
= (1-cosh x – coshy – cosh z) -1
.
G1(1,1- γ2, 1- γ3, zcosh -ycosh cosh x-1
cosh,
zcosh -ycosh cosh x-1
cosh
zy
) (1.3.5)
When H1 and G1 are defined by Horn (1931)
G1(1,, ’; cosh x, cosh y) = !!
)'()()(
, nm
nmmnnm
onm
coshm x cosh
n y (1.3.6)
H1(,,γ,δ; cosh x, cosh y) = m
nmnnm
onm nm )(!!
)()()(
,
cosh
m x cosh
n y (1.3.7)
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We also expand (1-cosh x- t
z
t
y
1
coshcosh
)-1
by taking = (1-cosh x) -1
as factor but that will only reduce FE to F4.
Similarly, we have FG(1, 1, 1, 1, 2, 3; γ1, γ2, γ3; cosh x, cosh y, cosh z)
= pnm
pnmpnm
onm nm
)()(!!
)()()()(
21
3211
,
coshm x cosh
n y cosh
p z (1.3.8)
absolutely convergent if m + n = 1 and m + p = 1 while
|cosh x|<1, |cosh y|<1, and |cosh z|<1
Its Integral representation is given by S. Saran (1955)
FG=
c
tti
1)1()()2(
)2()()(2
11
2F1(,1,:γ1: t
xcosh
)F1 (1,; 2;2;γ2; t
z
t
y
1
cosh
1
cosh
)dt (1.3.9)
2F1(,, 2;cosh x)=(1- 2
1
cosh x) -
2F1
2
2
)cosh2(
)(cosh;
2
1,
2
1
2
1,
2
1
x
x
(1.3.10)
absolutely convergent if m + n = 1 and m + p = 1 while |cosh x| <1, |cosh y|< 1, and |cosh z|< 1
It’s integral representation is given by
2F1 (,,’, +’; cosh x, cosh y)
(1-cosh y) -
2F1(,,+’; y
yx
cosh1
coshcosh
) (1.3.11)
using (1.3.10) and then taking the new variable of integration u given by t = 1/2cosh x + (1-1/2 cosh x) u in FG’s integral we will get
FG(γ1= 21)=(1-
x2
1Cosh
)-1
c
uui
1)1()()2(
)2()()(2
11
2F1(22
2
)cosh2(
)(cosh;
2
1;
2
1
2
1,
2
1
ux
x
) (1.3.12)
F1(,2, 3, γ3;(
)1)(2
cosh1(
cosh,
)1)(2
cosh1(
cosh
ux
z
ux
x
)du
Now using (1.3.11) after putting γ2 = 2 + 3 and introducing a new variable of integration v given by:
v = (1- cosh z -
)cosh2
1x
(1-v)/ (1 -
)cosh2
1x
(1.3.13)
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 6 ISSN 2250-3153
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We will get, on Integration
FG(1, 1, 1, 1, 2, 3,21, 2,+3, 2+ 3; cosh x, cosh y, cosh z)
= (1 – cosh z - 2
1
cosh x) -1
H4(1, 2,1+ 2
1
, 2,+ 3;
cosh x)2
1-zcosh -(1
zcosh -yCosh ,
2coshz)-coshx-4(2
xCosh2
2
) (1.3.14)
where
H4(,,γ,δ; cosh x, cosh y) = nm
nnm
onm nm )()(!!
)()( 2
,
cosh
m x cosh
n y (1.3.15)
We will now consider the integral S. Saran (1955) for Fk to show that
FK(1, 2, 3, γ1+ γ3 – 1, γ2, γ1+ γ3-1; γ1, γ2, γ3; cosh x, cosh y, cosh z)
= (1 – cosh y – coshy z) -2
(1.3.16)
H2(1-γ1, 2, 1, γ1+ γ3-1, γ3;
xx
cosh,1zcosh ycosh
cosh
)
= (1 – cosh x) -1
(1– coshy y) -2
H2(1-γ3, 1, 2, γ1+ γ3-1, γ1; 1ycosh
cosh,.
1ycosh
cosh
zz
) (1.3.17)
= (1-cosh x) -1
(1-cosh y – cosh z) -2
G2(1, 2, 1-γ1,1- γ3; zcosh -y cosh -1
cosh,
ycosh -1
cosh zz
) (1.3.18)
where H2 and G2 are defined by
H2(,,γ,δ; ; cosh x, cosh y) = m
nnmnm
onm nm )(!!
)()()()(
,
cosh
m x cosh
n y (1.3.19)
and
G2(,’,,’ cosh x, cosh y)=!!
)()()()(
, nm
nmmnnm
onm
coshm x cosh
n y (1.3.20)
FK(1, 2, 2, 1, 2, 1,21, 2, 3,; cosh x, cosh y, cosh z)
= (1 – 2
1
cosh x)-
1 (1- cosh y) -2
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H4 (1, 2, 1+ 2
1
, γ3;
)cosh1)(cosh2
11(
cosh,
)cosh2(4
cosh2
2
yx
z
x
x
) (1.3.21)
This leads to Erdelyi’s (1948) when y = 0
To prove these we know that
FK(1, 2, 2, 1, 2,1; γ1, γ 2, γ 3; cosh x, cosh y, cosh z)
pnm
npmm
onm pnm )()()(!!!
)())(()(
321
2121
,
, cosh
m x cosh
n y cosh
p z (1.3.22)
absolutely convergent if p = (1-m) (1-n) where
|cosh x|< 1, |cosh y|<1, and |cosh z|<1
FK(1, 2, 2, 1, 2, 1;γ1, γ 2, γ 3; cosh x, cosh y, cosh z)
=
c
tti
1)1()()2(
)2()()(2
11
2F1(,1;γ1+ 2
1
: t
xcosh
).F2(2,2,1,γ2, γ3; cosh y,)1(
cosh
t
y
)dt (1.3.23)
|1|>|cosh x| and |cosh z/1– t|< 1-|cosh y| along the contour and 1 = +1-1 putting γ2= 2 and γ3 = 1 we have
F2 (2,2,1, 2, 1; coshy, )1(
cosh
t
z
= (1-coshy -
2))1(
cosh
t
z
= (1- coshy – cosh z) -2
)1
.coshcosh1
cosh(
!
)( 2
t
t
zy
z
m
m
om
m (1.3.24)
= (1- coshy) -2
)1
.coshcosh1
cosh(
!
)( 2
t
t
zy
z
m
m
om
m (1.3.25)
and
2F1 (, 1, ; coshy,
2)cosh
1()cosh
t
x
t
x
= (1- cosh x) -1
mm
om tx
x
m)
1
1.
cosh1
cosh(
!
)( 2
(1.3.26)
Thus if 1 = γ1 + γ3 -1 and 2 = γ2 we will have
FK=
c
tti
3)1()()2(
)2()()( 1
2
3131
21 )t-1
zCosh cosh1()
t
yCosh 1(
y
dt (1.3.27)
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Now writing down the expansion (1.3.24) and
(1- t
ycosh
)-1
=
)cosh
(!
)( 1
t
x
m
m
om
m (1.3.28)
and the integrating term by term by term we will be table to prove (1.3.16) – (1.3.24) can also be proved by applying (1.3.24) and
(1.3.26) and then integrating term by term.
Application of (1.3.25) and (1.3.26) will prove (1.3.17) the proof of (1.3.21) is however, similar to (1.3.14).
We can also prove the following results for FN
FN(1,γ2, γ2, γ1+ k-1-2, γ1+k-1; γ1, γ2,; γ3,; cosh x, cosh y, cosh z)
= (1-cosh x) -1
(1-cosh y)- 2
H2(1-k, 1,k, γ1+k-1, γ2;
zx
xcosh,
1cosh
cosh
) (1.3.29)
= (1-cosh x) -1
(1-cosh y)- 2
G2(1,k,1- γ1,1-k; z
z
x
x
cosh1
cosh,
cosh1
cosh
) (1.3.30)
FN(1, 2, 2, 1, 2, 1,γ1, γ2, γ3,; cosh x, cosh y, cosh z)
= pnm
npmm
onm pnm
)()(!!!
)()()()(
21
2121
,
coshm x cosh
n y cosh
p z (1.3.31)
absolutely convergent if m+p = 1 and n = 1
where |cosh x|<1, |cosh y|<1, and |cosh z|<1 and
it’s integral representations
FN(1, 2, 2, 1, 2, 1,γ1, γ2, γ3,; cosh x, cosh y, cosh z)
=
c
tti
1)1()()2(
)2()()(2
11
2F1(, 1, γ1;
)cosh
t
x
F2(2, 2, 1,γ2;cosh y, )1(
cosh
t
z
dt (1.3.32)
Where 1= +1-1 and |t|>|coshx| and |1-t|>|cosh z| along the contour.
II. REDUCIBLE CASES FOR THE QARDRUPLE HYPERGEOMETRIC FUNCTION
2.1 INTRODUCTION
The various Hypergeometric function of four variables are studied earlier by H. Exton, H. Srivastava and many others.
Here all 21 functions are given below in the terms of a table.
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1 K1 (a,a,a,a;b,b,b,c;d,e1,e2,d; x,y,z,t)
= p!n!m!n)k!,m)(e,p)(ek(d,
tzyp)xn)(c,mkp)(b,nmk(a,
11
pnmk
2. K2 (a,a,a,a;b,b,b,c;d1,d2,d3,d4; x,y,z,t)
= p!n!m!p)k!,n)(d,m)(d,k)(d,(d
tzyp)xn)(c,mkp)(b,nmk(a,
4321
pnmk
3 K3 (a,a,a,a;b1,b1,b2,b2,;c1,c2,c2,c1; x,y,z,t)
= p!n!m!n)k!m,p)(ck,(c
tzyp)xn,m)((bk,p)(bnmk(a,
21
pnmk
21
4 K4 (a,a,a,a;b1,b1,b2, b2;c,d1,d2,c; x,y,z,t)
= p!n!m!n)k!,m)(d,p)(dk(c,
tzyp)xn,m)((bk,p)(bnmk(a,
21
pnmk
21
5 K5 (a,a,a,a;b1,b1,b2, b2;c1,c2,c3,c4; x,y,z,t)
= p!n!m!p)k!,n)(c,m)(c,k)(c,(c
tzyp)xn,m)((bk,p)(bnmk(a,
4321
pnmk
21
6 K6 (a,a,a,a;b,b,c1,c2;e,d,d,d; x,y,z,t)
= p!n!m!p)k!nmk)(d,(e,
tzyp)x,n)(c,m)(ckp)(b,nmk(a, pnmk
21
7. K7 (a,a,a,a;b,b,c1,c2,d1,d2,d1,d2; x,y,z,t)
= p!n!m!p)k!m,n)(dk,(d
tzyp)x,n)(c,m)(ckp)(b,nmk(a,
21
pnmk
21
8. K8 (a,a,a,a;b,b,c1,c2;d,e1,d,e2; x,y,z,t)
= p!n!m!n)k!,m)(e,n)(ek(d,
tzyp)x,n)(c,m)(ckp)(b,nmk(a,
21
pnmk
21
9. K9 (a,a,a,a;b,b,c1,c2,e1,e2,d,d; x,y,z,t)
= p!n!m!p)k!nm)(d,,k)(e,(e
tzyp)x,n)(c,m)(ckp)(b,nmk(a,
21
pnmk
21
10 K10 (a,a,a,a;b,b,c1,c2;d1,d2,d3,d4; x,y,z,t)
= p!n!m!p)k!,n)(d,m)(d,k)(d,(d
tzyp)x,n)(c,m)(ckp)(b,nmk(a,
4321
pnmk
21
11 K11 (a,a,a,a;b1,b2,b3,b4,c,c,c,d; x,y,z,t)
= p!n!m!p)k!n)(d,mk(c,
tzyp)x,n)(b,m)(b,k)(b,p)(bnmk(a, pnmk
4321
12 K12 (a,a,a,a;b1,b2,b3,b4,c1,c1,c2,c2; x,y,z,t)
= p!n!m!p)k!n,m)(ck,(c
tzyp)x,n)(b,m)(b,k)(b,p)(bnmk(a,
21
pnmk
4321
13 K13 (a,a,a,a;b1,b2,b3,b4,c,c,d1,d2; x,y,z,t)
= p!n!m!p)k!,n)(d,m)(dk(c,
tzyp)x,n)(b,m)(b,k)(b,p)(bnmk(a,
21
pnmk
4321
14 K14 (a,a,a,c3;b,c1,c1,b;d,d,d,d; x,y,z,t)
= p!n!m!p))k!nmk(d,
tzyn)xm,p)(ckp)(b,,n)(cmk(a, pnmk
13
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15 K15 (a,a,a,b5;b1,b2,b3,b4;c,c,c,c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)x,n)(b,m)(b,k)(b,p)(b,n)(bmk(a, pnmk
43215
16 K16 (a1,a2,a3,a4;b; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)xn,p)(an,n)(ak,m)(ak,(a pnmk
4321
17 K17 (a1,a2,a3,b1,b2;c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)x,p)(b,p)(bn,n)(ak,m)(ak,(a pnmk
21321
18 K18 (a1,a2,a3,b1,b2;c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)x,n)(b,n)(bm,p)(ak,m)(ak,(a pnmk
21321
19 K19 (a1,a2,b1,b2,b3,b4;c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)x,n)(b,m)(b,n)(bk,m)(ak,(a pnmk
32121
20 K20 (a1,a2,b3,b4;b1,b2, a2, a2;c,c,c,c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)xn,m)(a,k)(b,p)(b,n)(b,m)(b,k)(a,(a pnmk
2214321
21 K21 (a,a,b6,b5;b1,b2, b3, b4;c,c,c,c; x,y,z,t)
= p!n!m!p))k!nmk(c,
tzyp)x,n)(b,m)(b,k)(b,p)(b,n)(b,m)(bk(a, pnmk
432156
All results are having convergent conditions and restrictions on parameters including variables due to Exton (1972)
As far as known to me all the results investigation in this journal are new and interesting which have a wide range of application in the
field mathematical series
2.2 In this Section Quadruple hypergeometric function reduced to the hypergeometric function of one variable.
(1) Theorem : By specializing the parameters of K14, we obtain the following
F(1,a,a,1;b1, b2, b1, b2;b1, b2 + b1+ b2, b1+ b2, b1+ b2; x, y,z,t)
= (x + y - xy)-1
(z + t - zt)-1
.
[x 2F1(b1+p,a2+p;b1+b2+p+q;x)+y2F1(b2+q,a2+p;b1+b2+p+q;y)]
.[z2F1(b1,a2;b1+b2;z)+y2F1(b2,a2,b1+b2;t)] (2.2.1)
condition and restrictions are given in Exton (1972)
proof:
F(1,a2,a2,1;b1, b2, b1, b2,;b1+b2, b1+ b2, b1+ b2, b1+ b2; x, y,z,t)
=!!!!)(
)()()1()a()1(
21
212
0,,, qpnm
tzyx
bb
bb qpnm
qpnm
qnpmqpnm
qpnm
(2.2.2)
=!!!!)()(
)()()()()1()a(!
2121
22112
0,,, qpnm
tzyx
bbqpbb
bqbbpbm qpnm
nmnm
qnpmqpn
qpnm
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=!
)(
)(
)()( 2
21
21
0, n
pa
qpbb
yxqbpb n
nm
nm
nm
nm
!!
)1()(
)(
)()( 2
21
21
0, qn
a
qpbb
tzbb qp
nm
qp
qp
qp
(2.2.3)
by virtue of the result the to carlitz formula in the term
=
nm
nm
nm
nm
yx
)(
)()(
0,
= (x + y – xy)-1
n
n
n
m )(
x)( 1
0
+
n
n
n
n )(
y)( 1
0
We have equation (2.2.3) in the term
=(x + y - xy)-1 !
)(
)(
)(
)(
x)( 2
21
1
2
021
1
1
0 n
pa
qpbb
yqb
qpbb
pb n
n
n
n
nn
n
n
n
.(z + t - zt)-1
!
)(
)(
)(
)(
)( 2
21
1
2
021
1
1
0 p
a
bb
tb
bb
zb p
p
q
p
pp
p
p
p
(2.2.4)
= (x+y-xy)-1
(z+t-zt)-1
!)(
y)()(
!)(
x)()(x
21
22
021
21
0 nqpbb
paqby
nqpbb
papb
n
n
nn
nn
n
nn
n
.
!)(
)()(
!)(
)()(
21
22
021
21
0 pbb
tabt
pbb
zabz
p
q
pp
pp
p
pp
p
then theorem follows
F(1,a2,a2,1;b1, b2, b1, b2,;b1+b2, b1+ b2, b1+ b2, b1+ b2; x, y,z,t)
=(x + y - xy)-1
(z + t -zt)-1
[x2F1(b1+p,a2+p;b1+b2+p+q;x)+y2F1(b2+q,a2+p;b1+b2+p+q;y)].
[z2F1(b1,a2;b1+b2;z)+y2F1(b2,a2,b1+b2;t)]
The completes the derivation of (2.2.1).
(2) Theorem by specializing the parameters of K14, we obtain the following.
(i) F(1,a2,a2,1;b1, b2, b1, b2,;b1+b2, b1+ b2, b1+ b2, b1+ b2; 1, 1,1,1)
=
)()(
)()(
)()(
)()(
2211
2221
2212
2221
apbbqb
aqpbqpbb
aqbbqb
paqbqpbb
)()(
)()(
)()(
)()(
2211
2221
2212
2221
abbb
abbb
abbb
abbb
(2.2.6)
(ii) F(1,a2,a2,1;a2,a2,a2,a2;2a2,2a2,2a2,2a2;1,1,1,1)
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=
)()(
)2(2
)()(
)()(
)()(
)()2(
22
2
22
2
22
2
aa
a
qaqa
pqqpa
qaqa
pqqpa
(2.2.7)
(iii) F(1,1,1,1;1,1,1,1;2,2,2,2;1,1,1,1)
= )1()1(
)2(
pq
qp
[(q+1)+ (p+1)] (2.2.8)
(1) Proof :-
now putting x = y = z = 1 in equation (2.2.1) than we get
F(1,a2,a2,1;b1,b2,b1,b2;b1+b2,b1+b2,b1+b2,b1+b2;1,1,1,1)
=[2F1(b1+p,a2+p;b1+b2+p+q;1)+2F1(b2+q,a2+p;b1+b2+p+q;1)].
[2F1(b1,a2;b1+b2;1)+ 2F1(b2,a2,b1+b2;1)] (2.2.9)
now Apply Gauss’s summation theorem (1812)
2F1(,,γ;1)= )()(
)()(
(2.2.10)
using equation (2.2.10) in (2.2.9) we get
=
)()(
)()(
)()(
)()(
2211
2121
2212
2221
apbbqb
aqpbqpbb
aqbbqb
paqbqpbb
)()(
)()(
)()(
)()(
2211
2221
2212
2221
abbb
abbb
abbb
abbb
(2.2.11)
This completes the derivation of (2.2.6)
(2) Proof:- now putting b1 = b2 = a2 in equation (2.2.11)
F(1,a2,a2,a21;a2,a2,a2,a2;2a2,2a2,2a2,2a2;1,1,1,1)
=
)()(
)2(
)()(
)2(
)()(
)()(
)()(
)()2(
22
2
22
2
22
2
22
2
aa
a
aa
a
papa
qpqpa
qaqa
pqqpa
=
)()(
)2(
)()(
)()(
)()(
)()2(
22
2
22
2
22
2
aa
a
papa
qpqpa
qaqa
pqqpa
(2.2.12)
This competes the derivation of (2.2.7)
(3) Proof: now putting b1 = b2 = a2 = 1 in equation (2.2.12)
F(1,1,1,1;1,1,1,1;2,2,2,2;1,1,1,1)
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 13
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=
)2()1(2
)2(
)1(2
)2(
q
qp
q
qp
= )1p()1(
)2(
q
qp
[(q+1)+(p+1)] (2.2.13)
This completes the derivation of (2.2.8)
2.3 In this Section Quadruple hypergeometric function reduced to the Appell hypergeometric function
(1) Theorem By specializing the parameters of K12,K10,K15 we obtain the following
K12 (a, a, a, a; b1, b2, b3, b4, c1, c1, c2, c2; x,y,z,t)
= F1 (a, b1, b2; c1; x, y) F1 (a+m+n, b3, b4 ; c2 ; z,t) (2.3.1)
K12 (a, a, a, a; b1, b2,b3,b4, c1,c1, c2,c2; 1,1,1,1)
= )()(
)()(.
)()(
)()(
4322
4322
2111
2111
bbcac
bbnmacc
bbcac
bbacc
(2.3.2)
K10 (a, a, a, a; b,b,c1,c2 ; d1, d2, d3, d4; x, y, z, t)
= F4 (a, b; d1, d2; x, y) F2 (a+m+n, c1,c2;d3, d4; z, t) (2.3.3)
K15 (a, a, a, b5; b1, b2, b3, b4; c, c, c, c; x, y, z, t)
= F1 (a, b1, b2; c; x, y) F3 (a+m+n, b5, b3, b4; c+m+n; z, t) (2.3.4)
where (F1, F2, F3, F4,) are Appell hypergeometric function of two variables.
(1) Proof:
Now Quadruple hypergemotric function can be reduced to Appell function of two variable.
K12 (a, a, a, a; b1, b2, b3, b4, c1, c1, c2, c2; x, y, z, t)
=!!!!!!)(
)()()()()(
1
4321
0,,, qpnm
tzyx
nmc
bbbba qpnm
nm
qpnmqpnm
qpnm
(2.3.5)
=!!
)()(
)(
)(
!!)(
)()()( 43
21
21
0,,, qp
tzbb
c
nma
nmc
yxbba qp
qp
qp
qp
nm
nm
nmqpnm
qpnm
=
!!)(
)()()(
!!)(
)()()(
2
43
0,1
21
0, qpc
tzbbnma
nmc
yxbba
qp
qp
qpqp
qpnm
nm
nmqpnm
nm
= F1 (a, b1, b2; c ; x, y) F1 ( a+m+n,b3,b4; c2; z, t) (2.3.6)
where F1 is Appell function of two variable
This completes the derivation of (2.3.1)
(2) Proof :- now putting x = y = z = t= 1 in equation (2.3.6)
K12 ( a, a, a, a; b1, b2, b3, b4, c1, c1, c2, c2; 1,1,1,1 )
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= F1(a,b1,b2;c1; 1,1) F1 ( a+m+n,b3,b4; c2; 1,1) (2.3.7)
Now Apply Gauss's summation theorem
F1(,,1,γ;1,1)=)()(
)()(
1
1
(2.3.8)
Using equation ( 2.3.7) and (2.3.8)
= )()(
)()(.
)()(
)()(
4322
4322
2111
2111
bbcac
bbnmacc
bbcac
bbacc
(2.3.9)
This completes the derivation of (2.3.2)
(3) Proof:
K10 (a, a, a, a; b, b, c1, c2; d1, d2, d3, d4; x, y, z, t)
=!!!!)()()()(
)()()()(
4321
21
0,,, qpnm
tzyx
dddd
ccba qpnm
qpnm
qpnmqpnm
qpnm
(2.3.10)
=!!
)(
)()(
)()(
!!)()(
)()( 2
43
1
210,,, qp
tzc
dd
ca
nmdd
yxbaqp
q
qp
pqp
nm
nm
nmnm
qpnm
=
!!)()(
)()()(
!!)()(
)()(
43
21
0,210, qp
tz
dd
ccnma
nmdd
yxba qp
qp
qpqp
qpnm
nm
nmnm
nm
= F4 ( a, b; d1, d2 ; x, y) F2 ( a+m+n, c1,c2; d3, d4; z,t) (2.3.11)
where F4 & F2 are Appell function of two variable
This completes the derivation of (2.3.3)
(4) Proof:
K15 (a, a, a, b5; b1, b2, b3, b4; c, c, c, c; x, y, z, t)
=!!!!)(
)()()()()()( 43215
0,,, qpnm
tzyx
c
bbbbba qpnm
qpnm
qpnmqpnm
qpnm
(2.3.12)
=!!
)(
)(
)()()(
!!)(
)()()( 34521
0,,, qp
tzb
nmc
bbnma
nmc
yxbbaqp
p
qp
qqp
nm
nm
nmnm
qpnm
=
!!
)(
)(
)()()(
!!)(
)()()( 345
0,
21
0, qp
tzb
nmc
bbnma
nmc
yxbbaqp
q
qp
qqp
qpnm
nm
nmnm
nm
= F1( a, b1, b2; c ; x, y) F3(a+m+n, b5,b3,b4; c+m+n; z, t) (2.3.13)
Where F1 & F3 are Appell function of two variable
This completes the derivation of (2.3.4)
2.4 In this Section Quadruple hypergeometric function reduced to the Appell hypergeometric function Lauricella's set
and Saran
(1) Theorem By specializing the parameters of K2, K12, K15 , K6 we obtain the following
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K2 ( a, a, a, a; b, b, b, c; d1 d2, d3, d4; x,y,z,t)
= FE ( a, a,a, b,b,c,; d1, d2,d3 ; x,y,t) 2F1( a+m+n+q,b+m+n, d4; z ) (2.4.1)
K2, (a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y,l,t)
)()(
)22()(
44
44
nmbdqnmad
nmqbadd
FE (a,a,a,b,b,c,; d1,d2,d3; x, y, t) (2.4.2)
K12 ( a, a, a, a; b1,b2, b3,b4, c1, c1, c2, c2; x,y,z,t)
= FG( a, a, a, b1,b3, b4, c1,c2, c2; x,z,t) 2F1 (a+m+p+q,b2;c1+m;y) (2.4.3)
K12 ( a, a, a, a; b1, b2, b3, b4, c1, c1,c2, c2 ; x,l,z,t)
)()(
)()(
121
211
qpacbmc
qpbacmc
FG ( a, a, a, b1,b3, b4,; c1, c2, c2; x, y, z, t ) (2.4.4)
K15 ( a, a, a, b5;b4, b1,b2,b3; c, c, c,c; x, y, z, t )
= FS( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 (a+m+n,b3 ;c+q+m+n; z) (2.4.5)
K15 ( a, a, a, b5; b4, b1, b2, b3;c, c, c, c; x, y, z, t)
)()(
)()(
3
3
aqcbnmqc
baqcnmqc
FS(b5, a, a, b4, b1, b2,; c,c,c; x,y,t) (2.4.6)
K6 (a,a,a,a; b,b, c1,c2;e, d,d,d;x,y,z,t)
= FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; t) (2.4.7)
K6 ( a,a,a,a; b,b, c1,c2;e, d,d,d;x,y,z,t)
=)()(
)()(
2
2
cnpdmad
camdnpd
FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) (2.4.8)
Where (F4,F14, F8,F7) & (FE, FF, FG,FS) are Lauriceila's set & Saran Triple hypergeometric Series.
(1) Proof:- Now Quadruple hypergeometric function can be reduced to Lauricella's set and Saran Triple hypergometric Series.
K2 ( a,a,a,a; b,b,b,c; d1d2,d3,d4;x,y,z,t)
=!!!!)()()()(
)()()(
43210,,, qpnm
tzyx
dddd
cba qpnm
qpnm
qpnmqpnm
qpnm
(2.4. 9)
=!
)(
)(
)(
!!)()()(
)()()(
34210,,, p
znmb
d
qnma
nmddd
tyxcba p
p
p
p
qnm
qnm
qnmqnm
qpnm
=
!)(
))((
!!!)()()(
)()()(
304210,, p
z
d
nmbqnma
qnmddd
tyxcba p
ppqnm
qnm
qnmqnm
nmq
= FE ( a, a, a, b, b,c,; d1 , d2, d3 , x,y, t) 2F1 ( a+m+n+q, b+m+n, d4; z) (2.4.10)
This completes the derivation of (2.4.1)
(2) Proof:
If z = 1, in euqation (2.4.10)
K2 (a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y, l,t)
= FE( a, a, a, b, b,c,; d1, d2, d3, x,y, t) 2F1 ( a+m+n+q, b+m+n, d4; 1) (2.4.11)
Now Apply Gauss's summation theorem in equation ( 2.4.11)
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F1(,,γ;1)= )()(
)()(
K2 ( a, a, a, a; b, b, b, c; d1, d2, d3, d4; x,y, l,t)
=)()(
)22()(
44
44
nmbdqnmad
nmqbadd
FE( a, a, a,b,b, c,; d1,d2,d3; x,z,t) (2.4.12)
This completes the derivation of (2.4.2)
(3) Proof: K12 ( a, a, a, a; b1, b2, b3,b4,c1,c2, c2 ; x,y, z,t)
=!!!!)()(
)()()()()(
21
4321
0,,, qpnm
tzyx
cc
bbbba qpnm
qpnm
qpnmqpnm
qpnm
(2.4.13)
=!)(!!!
)()(
)()(
)()()()(
1
2
21
431
0,,, nmcqpm
ybqpma
cc
tyxbbba
n
n
nn
qpm
qnm
ppmqpnm
qpnm
=
!)(
)()(
!!!)()(
)()()()(
1
2
021
431
0,, nmc
ybqnma
qpmcc
tyxbbba
n
n
nn
pqpm
qpm
qpmqnm
qpm
FG ( a, a, a, b1, b3, b4; c1, c2,c2; x, z, t) 2F1 (a+m+p+q; b2; c1+m; y) (2.4.14)
This completes the derivation of (2.4.3)
(4) Proof:
When y = 1, in equation (2.4.14) Then
K12 ( a, a, a, a; b1, b2, b3, b4, c1, c1, c2, c2; x,I, z,t)
=FG (a, a, a, b1, b3, b4; c1, c2,c2; x, z, t). 2F1 (a+m+p+q; b2; c1+m; 1) (2.4.15)
Now Apply Gauss's summation theorem in equation ( 2.4.15)
F1(,,γ;1)= )()(
)()(
=)()(
)()(
121
211
qpacbmc
qpbacmc
FG( a, a, a,b1,b3, b4,c1, c2, c2,; x,z,t) (2.4.16)
This completes the derivation of (2.4.4)
(5) Proof :- K15 (a, a, a, b5; b1, b2, b3, b4 ; c, c, c, c; x; y, z, t)
=!!!!)(
)()()()()()( 43215
0,,, qpnm
tzyx
c
bbbbba qpnm
qpnm
qpnmqpnm
qpnm
=!)(!!!)(
)()()()()()()( 32145
0,,, pnmqcnmqc
zbnmayxtbbbab
pnmq
q
pp
nmq
nmqnmq
qpnm
=
!)(
)()(
!!!)(
)()()()()(2
0
2145
0,, nnmqc
ybqppma
pmqc
yxtbbbab
n
n
nn
pnm
nmq
nmqnmq
qpm
= FS( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 ( a+m+n,b3 ;c+q+m+n; z ) (2.4.17)
This completes the derivation of (2.4.5)
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(6) Proof :- When z = 1 in equation (2.4.17) Then
K15 (a, a, a, b5, b4, b1, b2, b3; c, c, c, c; x, y, 1, t )
= Fs( b5, a, a, b4,b1,b2;c,c,c; x,y,t) 2F1 ( a+m+n,b3 ;c+q+m+n;1) (2.4.18)
Now Apply Gauss's summation theorem in equation
F1(,,γ;1)=)()(
)()(
=)()(
)()(
3
3
aqcbnmqc
baqcnmqc
FS(b5,a, a,b4,b1, b2,c, c, c,; x,z,t) (2.4.19)
This completes the derivation of (2.4.6)
(7) Proof:- K6 ( a, a, a, a; b, b, c1,c2 ; e, d, d, d; x, y, z, t)
=!!!!)()(
)()()()( 21
0,,, qpnm
tzyx
de
ccba qpnm
qpnm
qpnmqpnm
qpnm
(2.4.20)
=!!!!
)()(
)()(
)()()( 21
0,,, qpnm
tcnpma
de
yzxcba q
qpnm
npm
pnmnpm
qpnm
=
!)(
)()(
!!!)()(
)()()( 2
0
1
0,, qnpd
tcnpma
npmde
yzxcba
q
q
pnpm
npm
pnmnpm
qpm
= FF( a, a, a,b, c1, b, ;e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; t) (2.4.21)
This complete the derivation of (2.4.7)
(8) Proof :-
When t = 1 in equation (2.4.21) Then
K6 ( a, a, a, a; b, b, c1,c2; e, d, d, d; x, y, z, 1)
= FF( a, a, a,b, c1, b,; e,d,d; x,z,y) 2F1 ( a+m+p+n,c2 ;d+p+n; 1) (2.4.22)
Now Apply Gauss's summation theorem in equation
F1(,,γ;1)= )()(
)()(
=)()(
)()(
2
2
cnpdmad
camdnpd
FF(a, a,a,b,c1, b,;e,d,d; x,z,y) (2.4.23)
This completes the derivation of (2.4.8)
REFERENCES
[1] APPELL, P. (1880): Surles series hypergeometric de deux variable, et surds equationa diiferentiells linearies aux derives partielles, C.R. Acad. Sci. Paris, 90, 296-298.
[2] ERDELYI, A. (1948) : Transformation of the hypergeometric functions of four variables, Bull soco grece (N.S.) 13, 104-113.
[3] EXTON, H. (1972) : Certain hypergeometric functions of four variables, Bull soco grece (N.S.) 13, 104-113.
[4] HORN, J. (1931) : Hypergeometric Funktionen Zweier Veranderlichen Math. Ann, 105, 381 – 407.
International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 18
ISSN 2250-3153
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[5] SARAN, S. (1955) : integrals associated with hypergoemetric functions of there variables, Not. Inst. of Sc. of India, Vol. 21, A. No. 2, 83-90.
[6] SARAN, S.(1957): Integral representations of laplace type for certain hypergeometric functions of three variables, Riv. Di. Mathematica, parma 133-143.
[7] WHITTAKER, E.T. AND WATSON, G.N. (1902) : A course of Modern Analysis
AUTHORS
First Author – Pooja Singh, Research Scholar, Department of Mathematics, NIMS University, Shobha Nagar, Jaipur (Rajasthan)
Second Author – Prof. (Dr.) Harish Singh Department of Business Administration, Maharaja Surajmal Institute, Affiliated to Guru
Govind Singh Indraprastha University, New Delhi. [email protected]
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