ETNA
Electronic Transactions on Numerical Analysis.
Volume 25, pp. 115-120, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
AN INTEGRAL REPRESENTATION OF SOME HYPERGEOMETRIC
FUNCTIONS
K. A. DRIVER AND S. J. JOHNSTON Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. The Euler integral representation of the Gauss hypergeometric function is well known and plays
a prominent role in the derivation of transformation identities and in the evaluation of , among other
applications. The general hypergeometric function has an integral representation where the integrand
involves . We give a simple and direct proof of an Euler integral representation for a special class of functions for !#" . The values of certain $% and &'$ functions at (*) , some of which can be derived using
other methods, are deduced from our integral formula.
Key words. 3F2 hypergeometric functions, general hypergeometric functions, integral representation
AMS subject classification. 15A15
1. Introduction. The general hypergeometric function is defined by
+,.-0/21%3547686768410+9:73;486767684:-9<=?>
@
/F1%38= B 8
6 6767/21G+H= B
BCED /2: 3 = B 686768/: - = B
A
B
<
IKJ 4ML <.L'NPO
where
/FQ.= B >SR
QT/FQVUWO5=X676867/FQYU
I[Z
I]\
O;=
4
O
I
4
OH4
>_^`4.Q_>_
a ^
is Pochhammer’s symbol. The Euler integral representation of the Gauss hypergeometric
function, or b ,.3 function, is well known in the literature (cf. [1], [3]) and is formulated as
follows (cf. [1], p. 65, Theorem 2.2.1):
T HEOREM 1.1. If Re /2c7=ed Re /2:7=edf^ , then
b ,.3G/F14:59c9<X=g>
h
/2:=
/Fc7=
h
h
/2c
3
Z
3
Z
/O
:=gi Dkjlm
j
=n
3
Z
/O
mKlm
<
=
j K
m op0j
in the < plane cut along the real axis from 1 to q . Here, it is understood that arg j >
Z
Z
=g>k^ and /O
< =
arg /O
has its principle value.
j
j mKo
This integral may be viewed as the analytic continuation of the b , 3 series for Re /Fc8=rd
Re /2:7=Tds^ and it yields the famous Gauss summation formula of 1812, namely
Z
Z
1
:=Tdt^ , we have
T HEOREM 1.2 (cf. [1], p. 66, Theorem 2.2.2). For Re /Fc
b ,.3G/F14:9c98O;=g>
A
@
u CKD
/21= u
/2:= u
J
/2c7=u[v
w
/2c7=
h
>
h
/2c
Zh
/Fc
1'=
Z
h
1
/Fc
Z
Z
:=
:=
Received April 21, 2005. Accepted for publication October 31, 2005. Recommended by I. Pritsker.
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University
of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa (kathy@maths.wits.ac.za).
This material is based on research supported by the National Research Foundation under Grant Number 2053730.
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
(sjohnston@maths.wits.ac.za).
115
ETNA
Kent State University
etna@mcs.kent.edu
116
b , 3
K. A. DRIVER AND S. J. JOHNSTON
The case where one of the numerator parameters is a negative integer, thereby making the
a finite sum, is known as the Chu-Vandermonde identity (cf. [1], p. 67, Corollary 2.2.3)
Z
b , 3 /
Z
/Fc
vx419c97O5=x>
1'= u
6
/Fc7=u
\S|
The values of -yz3 , - functions at 1 with {
encompass the beautiful identities due
to Pfaff-Saalschutz, Dougall, Dixon, Rogers, Ramanujan, Whipple and other authors. For
further discussion, see [1], Chapters 2 and 3.
2. Results. I In [3], Theorem 38, Rainville proves a general integral representation for
where is a positive integer and Re /FQ.=edf^ , Re /F}z=eds^ , namely
+7y B , -y B
yz3
y B 3
4g10+'4* B *
4  B 4
6867684
 B
yXH
„ y…3
yK„Hy m B
yX„
: 3 4‚67686€4ƒ: - 4  B 4  B
4†67686‡4 
B
1'34€67686
+7y B , -y B~
3
„

3
Z
< 
/
‰j m m
D Š
m
j
/2Qƒ4}z= i‹
>
„
<=
3
B ˆ
1 3 4€67686€4g1 +
9c<
: 3 4‚67686Œ4ƒ: -
+0,…- ~
m
3 9c
m
p
B5ˆ
j
<z6
We begin with a direct proof of an
Euler yz
integral
representation for the special class of hypery…3
3
geometric functions  , bŽ 14 b l 4 l b 9 b n 4 n b 9< .
T HEOREM 2.1. For Re /Fc7=Td Re /2:7=edf^ ,
(2.1)
ˆ
: :gU_O c cƒU_O
9 | 4
9<
|
|
 , b ~ 14 | 4
>
/2:=
h
/Fc7=
h
/Fc
h
3
Z
3
/O
j lm
:=ƒi D
Z
3
= n
j
mKlm
Z
/O
<
j
b =
Proof. Suppose that L <.L'N‘O . Then the left hand side of (2.1) becomes
@
y 3
z
/21= B Ž b  B Ž b  B
l z
y 3 l
I J
X
BCED Ž b n  B Ž n b  B
A
(2.2)
B
<
@
/F1'= B /2:= b B
IXJ
BCKD /Fc7= b B
A
>
B
<
since (cf. [3], p. 22)
| b B“’ Q
/FQ.= b B >
Q•UWO ˆ
|
~
|ƒ” B
6
B
Now
/2:7= b B
/Fc8= b B
>
/:=
h
/2c7=
h
Z
/Fc
h
:=
|HI
/:xU
h
h
=
/2cgU
/Fc
h |HI
Z
:7=
>
=
h
/Fc7=
h
/2:=
h
‰
Z
/Fc
|HI
/2:xU
:=
4c
Z
:=6
Therefore, for Re /Fc8=Td Re /:=Tdt^ , L <…L%NkO , the right hand side of (2.2) becomes
/2:=
h
>
h
/Fc7=
h
/Fc
>
h
/2:=
Z
:=
Z
/Fc7=
h
h
/2c
BCKD
B
/F1'= B <
IXJ
3
Z
3
:=–i D‘jlm
3
i D
j l
Z
/O
:=–i D‘jlm
/2c
h
@
3
/Fc7=
h
/2:=
h
A
3
/O
y b B
j
=n
Z
j
=n
3
m
3 A
mXlm
j
@
3
= n
mXlm
/F1'= B
BCKD
3
mXlm
Z
/O
/O
Z
<
j
b =
p0j
B
F/ < b =
j
IXJ
mKo5p0j
6
pHj
mXo pHj
6
ETNA
Kent State University
etna@mcs.kent.edu
117
AN INTEGRAL REPRESENTATION OF SOME HYPERGEOMETRIC FUNCTIONS
This proves the result for L <…L.N—O . Since the integral is analytic in the cut plane, the result
holds for < in this region as well.
d ^ ,
C OROLLARY 2.2. For Re /Fc7=Td Re /2:7=ef
ˆ
: x
: UWO c cxU_O
Z
vx4 | 4
9 | 4
9<
|
|
 , b ~
>
/2:7=
h
3
/Fc7=
h
Z
/2c
h
Z
T HEOREM 2.3. If Re /2c7=ed Re /2:7=edf^ and Re /Fc
(2.3)
ˆ
: x
: UWO c cxU_O
9 | 4
98O
|
|
 , b ~ 14 | 4
/Fc7=
h
>
Z
:=
Z
1
/Fc
h
:=
Z
3
=n
j
:=Tdt^
Z
/2c
Z
/O
jlm
1
Zh
/Fc
h
3
:=?i D
mXlm
Z
<
j
b = u
pHj
, then
b , 3 /F14:59c
1=
/O
Z
19
Z
O;=6
Proof. Putting <•>˜O in equation (2.1), the left hand side of (2.3) becomes
3
Z
/O
:7=™i Dkjlm
h
3
/Fc8=
3
h
/ O
Z
/:= /Fc
:7=ši DPjlm
h
h
@
Z
A
1 ˆ
/Fc8=
~
h
Z
I
/:= /Fc
:7= BCKD
h
h
@
Z
A
/Fc8=
1 ˆ
~ I
h
Z
/:= /Fc
:7= BCKD
h
h
/2:7=
h
>
>
>
B  >œ
Now Ž K
m
m o
3
/Fc8=
h
Z
/2c
3
=n
Z
=
/O?U
=
j X
m o
j mXo5pHj
@
Z
1 ˆ B
3 A
Z
~ I
= n
j mKlmXo;m BCKD
j pHj
3
y B 3
3
Z
/O
= n
j mXlmXo5m pHj
m
i D j l
j
/O
mKlm
I
/2:gU
h
=
/Fc
h
Z
/Fc
Zh
I
1›U
Z
:
1'=
6
=
y B ž
and /2Q…= B >S¡%œ  ž . Therefore we have
3žFŸ ž Ÿ
B7 œ
o
/2:=
h
/Fc7=
h
/Fc7=
h
>
Z
/Fc
h
/Fc
h
/Fc7=
>¢h
/Fc
h
A
:=
Zh
/2c
1'=
h
h
:
/2c
Z
1 ˆ
I
Z
:=
Z
=
/2c
h
Z
/
O;=
BCED
1=
Z
I
/:xU
h
@
A
1=
/2c
Z
~
Z
:
1'=
Zh
¡'œ 
BCKD
Z
/2c
@
B
I
1“U
/21'= B
IKJ
Z
19
Z
Z
:
1=
=
/2:= B
Z
1'= B
/2c
b , 3 /F14:59c
:=
Z
/2c
Zh
O;=6
R EMARK 1. An alternative proof of Theorem 2.3 can be found in [4], formula (3.7).
C OROLLARY 2.4. If Re /Fc7=ed Re /:=edf^ , then
ˆ
: :gUWO c cƒU_O
Z
vx4 | 4
9 | 4
98O
|
|
 , b ~
Z
/2c
>
:= u
b ,.30/
/Fc8=u
Z
vx4:59cgU‹vx9
Z
O5=6
Generalising Theorem 2.1 in the obvious way yields the following result.
T HEOREM 2.5. If Re /2c7=ed Re /2:7=edf^ , then
-y…3 , - ~ 14
>
h
:
4
{
/2:7=
:gUWO
{
h
/Fc7=
h
/2c
4867686
:xU
:=?i D
O
9
{
3
Z
Z
{
3
j lm
/O
c
{
4
Z
j
= n
cƒUWO
{
3
mXlm
cgU
486768674
/O
Z
Z
{
<
{
O
j
=
mXo pHj
6
9<
ˆ
6
ETNA
Kent State University
etna@mcs.kent.edu
118
K. A. DRIVER AND S. J. JOHNSTON
Proof. We use (cf. [3], p. 22, lemma 6)
/FQ.=- B >
- B
{
Q ˆ
~
B
{
QYUWO ˆ
~
B
{
Q•U
67676 ~
Z
{
O ˆ
B
{
and the proof proceeds as in Theorem 2.1.
It follows from Theorem 2.5 with { >_£ that for Re /Fc8=Td Re /2:=eds^ ,
:
¤ ,  ~ 14
>
£
/Fc7=
Z
/2c
h
|
:xU
4
£
h
/2:7=
h
:xU_O
4
c
9
£
£
3
cgUkO
4
3
£
Z
/O
:=?i DPjlm
ˆ
9<
£
3
=n
j
|
cgU
4
Z
/O
mXlm
<
 =
j
mXo5pHj
and this leads to the following Z result.
Z
1
:=Tdf^ ,
T HEOREM 2.6. For Re /Fc
:
¤ ,  ~ 14
:.U_O
4
£
£
/2c7=
>¢h
/2c
h
Zh
1'=
Z
:
£
1'= A
Z
/Fc
h
c
9
£
Z
/Fc
|
:.U
4
cgU_O
4
£
@
:= BCKD
/F1'= B /
IKJ
|
cxU
4
£
Z
Z
/Fc
ˆ
98O
B
O5=
/:= B
1= B
Z¥I
b , 3 /
I
4:…U
Z
9c
1¦U
I
9
Z
O;=6
Proof. Letting { >W£ and <Y>§O in Theorem 2.5, we have
:
¤ ,  ~ 14
>
£
>
>
>
h
/2:=
>
>
>
/2:=
h
/Fc7=
h
>
/Fc
h
/Fc7=
>¢h
h
/Fc
Z
Z
/Fc
h
Z
:= BCED
/2c7=
h
h
Zh
Zh
Z
/Fc
h
/Fc
A
Z
¨A
:= BCED
Z
Z
:
1'=
1'=
@
/Fc
Z
h
h
:
/Fc
Z
1'= A
Z
/O
=n
j
Z
/O
j
=n
Z
/21= B / O5=
IKJ
B
B
CKD
B
CKD
@
A¨
:= BCKD
Z
=n
j
Z
/21= B / O5=
IKJ
1'= A
Z
Z
@
:= BCKD
A¨
3
3
mKlmKo;m
@
3
B y
3
B
Z
Z
j
3
I©
ˆ
I©
Z
/F1'= B / O5=
IXJ
B
Z
/F1'= B / O5=
IXJ
B
ˆ
~
I©
¨
U
3
mXlmXo;m
/2c
I
=
¨ 1›U
/O
/Fc
h I
Z
Z ©
ˆ
/Fc
Z
/2:= B / :.U
Z
1'= B /2c
j
p0j
Z
pHj
1'=
=
¨
/:= y B
Z
/Fc
1'= y B
ˆ
¨
ˆ
mXlmXo;m
:
U
p0j
3
=n
j
B
I©
~
CKD
pHj
=
j
B
3 A¨
B
=
j
/O?U
mXlmXo;m
3
Z
/O?U
j
lm
©
h
I©
~
mXo5pHj
B
B
O;=
y
/2:xU
h
=n
B y
i DPj
~
= n
j
/O
lm
~
b =
j
/21'= B /
IXJ
/O
lm
i D‘j
U
Z
B y
3
mXoGp0j
j
BCKD
j
Z
/F1'= B / O5=
IXJ
CKD
/O?U
3 A
i D
 =
j
mKlmKo;m
B
CKD
B
Z
/O
mKlm
3
ˆ
98O
£
Z
/F1'= B / O5=
IXJ
B
|
cgU
4
£
B
¨A
:= BCED
/Fc
/Fc
@
A
/2c7=
h
3
3
@
A
/Fc
h
3
cƒU_O
/O
jlm
:= BCED
/2c7=
h
£
3
@
A
/2c7=
h
4
:=ƒi D‘jlm
/Fc
h
/2:=
h
:=ƒi D
/2c7=
h
/2:=
h
Z
/Fc
h
£
3
c
9
:=ƒi D‘jlm
3
/2c7=
h
/2:=
h
Z
|
:xU
4
£
/Fc
h
/2:=
h
:xU_O
/2c7=
h
/2:=
h
4
¨
I
=
1“U
¨
I
=
from [3], p. 65
ETNA
Kent State University
etna@mcs.kent.edu
119
AN INTEGRAL REPRESENTATION OF SOME HYPERGEOMETRIC FUNCTIONS
¨
B
@
B
©
Z
Z
Z
I
I
¨
ˆ
/2c7= /Fc
:
1'= A
/F1'= B / O5= /:= B ¨A
/2:gU
=
~
> h
ª
6
Zh
Z
IKJ
Z
Z
I
/2c
1'= /Fc
:= BCED
/Fc
1= B
/Fc
1“U
=
CKD
h
h
Now,
¨
B
¨A
I©
~
CKD
A¨
>
ˆ
Z
/
CKD
B
=
Z
/Fc
Z¥I
b , 3 /
¨
© ¥
Z I
O5= /
=
J
¨
Z¥I
/
CKD
>
¨
/2:gU
=
Z¨
I
/2c
1›U ¨ =
B
A¨
>
I
I
1¦U
I
4:xU
¨ /2c
I ¨
/:xU
/2:xU
Z
I
=
Z
I
since
=
/
I«Z
J >¬/
=
Z
¨
O;=
Z¥I
/
=
O5=
J
=
9c
=
1›U ¨
Z ©
/
¨
IXJ ©
¨
I
1›U
Z
9
O;=6
Hence,
:
¤ ,  ~ 14
/Fc7=
£
>¢h
Zh
/Fc
h
:xU_O
4
4
£
Z
/Fc
1'=
Z
:
Z
h
/Fc
|
:.U
9
£
c
cƒU_O
4
ˆ
98O
£
£
B
Z
B
1'= A
/21= / ;
O = / := B
IXJ
Z
:= BCKD
/Fc
1'= B
@
£
|
cgU
4
Z¥I
b ,.3G/
I
4:xU
Z
9c
I
1›U
9
Z
O;=
as stated.
y 3
z
yz3
As 3 another application of Theorem 2.1, we consider the value of  , b Ž 14 b l 4 l b 9 b n , n b 9< 
at <]> b and obtain the following result.
T HEOREM 2.7. If Re /2c7=ed Re /2:7=edf^ , then
: :xUWO c cxU_O O ˆ
9 | 4
9 |
|
|
 , b ~ 14 | 4
A
|
>
@
o BCKD
Proof. From Theorem 2.1 with <]>
b
3
Z
~
I
1 ˆ
Z
/Fc
:7= B
Z¥I
b , 3 /
/2c7= B
4:9cxU
I
,
: :xUWO c cƒUWO O ˆ
9 | 4
9 |
|
|
 , b ~ 14 | 4
/Fc7=
h
/2:=
h | h
o
/2:=
h | h
o
/2:=
h
h
|
o
/2:=
>
>
>
>
h
/Fc
h
h
h
/2:=
h
o h
h
|
>
h
/Fc
h
|
>
/Fc
/2:=
o h
h
/Fc
/Fc
/Fc
3
Z
/Fc8=
Z
/Fc8=
Z
/Fc8=
Z
/Fc8=
Z
/Fc8=
Z
3
i D‘jlm
:=
3
3
3
:=ƒi DPjlm
3
@
A
Z
~
A
@
:= BCKD
~
Z
=n
j
Z
=n
j
I
1 ˆ
3 ~
Z
O
mXlm
3
|¦Z
/
mKlm
j
3
@
3 A
mKlm
3
i D‘jlm
3
~
BCKD
Z
/O
j lm
|
b =
/O
= n
j
Z
j
=n
j
I
p0j
mKo!pHj
j
Z
b ˆ mXo
O
Z
Ž OƒU_/O
mKlm
3
i D
Z
=n
3
1 ˆ
I
:= BCKD
/O
=n
j
/O
3
:=ƒi DPjlm
j
Z
/O
:=ƒi DPjlm
3
Z
/O
1 ˆ
b = 
mXo
/O
3y B
mKlm
Z
j
b = B
/OTU
3y B A¨
mKlm
p0j
B
j
CED
~
p0j
B
=
pHj
I©
¨
ˆ
j
p0j
9
Z
O;=6
ETNA
Kent State University
etna@mcs.kent.edu
120
>
h
>
h
>
¨
|
o
K. A. DRIVER AND S. J. JOHNSTON
¨
B
@
3
|
Z
I©
A
A¨
ˆ
/2c7=
1 ˆ
y
3
3y B
Z
~ I
~
o h Z
/O
=n
j K
m lm
p0j
/2:7= /2c
: = BCKD CED
i D‘jl © m
h
B
@
© Z
|
Z
I©
I
A
A¨
ˆ
/2c7=
1 ˆ
/:xU
= /Fc
: U
x
=
~ I
~
o h Z
h
h
I
/2:7= /2c
:= BCKD CED
2
/
g
c
U
U
=
¨
h
h
B
@
Z
I©
Z
¨
B
¨A
A
ˆ /2:= /Fc
1 ˆ
:=
~ I
~
6
B
F
/
8
c
=
y
BCED CKD
¨
I
=
But since /2c7= y B >­/Fc7= B /FcƒU
(cf. [3], p. 65), we have that
: :xUWO c cxUkO O ˆ
9 | 4
9 |
|
|
 , b ~ 14 | 4
o
¨A
A
o
Z
~
BCED
I
1 ˆ
I
1 ˆ
¨
I©
~
Z
/2c
ˆ
/2:=
Z
/Fc
I
/Fc7= B /2cƒU
B
:= B A¨
/Fc7= B
CKD
~
I©
¨
:= B
=
ˆ
¨
/2:=
/FcƒU
¨
I
=
6
¨
B
CKD
@
Z
~
CKD
BCED
|
>
B
A¨
A
|
>
We know that
@
~
I©
ˆ
/2:7=
/2cgU
¨
I
>
=
Z¥I
b ,.30/
4:59c.U
I
9
Z
O5=
and hence
: :xU_O c cgU_O O ˆ
9 | 4
9 |
|
|
 , b ~ 14 | 4
A
|
>
o
@
BCKD
Z
~
1 ˆ
I
Z
/Fc
/2c7= B
:= B
b , 3 /
Z¥I
I
4:59cxU
9
Z
O;=6
C OROLLARY 2.8. If Re /Fc8=Td Re /:=eds^ , then
: :xU_O c cƒU_O O ˆ
9 | 4
9 |
|
|
 , b ~ 14 | 4
>
|
A
@
o BCKD
~
Z
1 ˆ
I
ˆ
Z¥I : :gU_O c cƒU_O
4 | 4
9 | 4
97O
|
|
 , b ~
6
Proof. From Corollary 2.4, we can see that
ˆ
Z¥I : :xUWO c cgUWO
4 | 4
9 | 4
97O
|
|
 , b ~
>
/Fc
Z
/2c7= B
:= B
b , 3 /
Z¥I
4:9cgU
I
9
Z
O;=
which along with Theorem 2.7 gives the result.
R EMARK 2. This corollary is a special case of a transformation given by Chaundy [2].
REFERENCES
[1] G. E. A NDREWS , R. A SKEY, AND R. R OY , Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.
[2] T. W. C HAUNDY , An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser., 14 (1943),
pp. 55–78.
[3] E. D. R AINVILLE , Special Functions, The Macmillan Co., New York, 1960.
[4] F. J. W. W HIPPLE , On series allied to the hypergeometric series with argument -1, Proc. London Math. Soc.,
30 (1930), pp. 81–94.