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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


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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)]


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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


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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)


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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.

International Journal of Scientific and Research Publications, Volume 2, Issue 10, October 2012 9

ISSN 2250-3153

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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,


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


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,


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


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


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


ISSN 2250-3153

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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)


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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)


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


(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)


(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


(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


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)


(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)


(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)


(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)


(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)



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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)


(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

qq

qpnm

npm

pnmnpm

qpnm

=

!)(

)()(

!!!)()(

)()()( 2

0

1

0,, qnpd

tcnpma

npmde

yzxcba

q

q

qq

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)


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.


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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)

[email protected]

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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