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16
Generalized Hypergeometric Functions & Meijer
G
-Function
Generalized Hypergeometric Functions
16.11
Asymptotic Expansions
16.13
Appell Functions
Β§16.12
Products
β
Keywords:
generalized hypergeometric functions
,
products
Notes:
See
ErdΓ©lyi
et al.
(
1953a
, Β§4.3)
. For (
16.12.1
) see
Bailey (
1928
)
. For (
16.12.2
) see
Clausen (
1828
)
. For (
16.12.3
) see
Chaundy (
1969
, ChapterΒ 12, Problem 12)
.
Permalink:
http://dlmf.nist.gov/16.12
See also:
Annotations for
Ch.16
16.12.1
F
1
0
β‘
(
β
;
a
;
z
)
β’
F
1
0
β‘
(
β
;
b
;
z
)
=
F
3
2
β‘
(
1
2
β’
(
a
+
b
)
,
1
2
β’
(
a
+
b
β
1
)
a
,
b
,
a
+
b
β
1
;
4
β’
z
)
.
β
Symbols:
F
q
p
β‘
(
a
1
,
β¦
,
a
p
;
b
1
,
β¦
,
b
q
;
z
)
or
F
q
p
β‘
(
a
1
,
β¦
,
a
p
b
1
,
β¦
,
b
q
;
z
)
: alternatively
F
q
p
β‘
(
π
;
π
;
z
)
or
F
q
p
β‘
(
π
π
;
z
)
generalized hypergeometric function
,
z
: complex variable
,
a
,
a
1
,
β¦
,
a
p
: real or complex parameters
and
b
,
b
1
,
β¦
,
b
q
: real or complex parameters
Referenced by:
(10.8.3)
,
Β§10.8
,
Β§10.8
,
Β§16.12
,
Erratum (V1.0.17) for Section
10.8
Permalink:
http://dlmf.nist.gov/16.12.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
Β§16.12
and
Ch.16
16.12.2
(
F
1
2
β‘
(
a
,
b
a
+
b
+
1
2
;
z
)
)
2
=
F
2
3
β‘
(
2
β’
a
,
2
β’
b
,
a
+
b
a
+
b
+
1
2
,
2
β’
a
+
2
β’
b
;
z
)
.
β
Symbols:
F
1
2
β‘
(
a
,
b
;
c
;
z
)
:
=
F
β‘
(
a
,
b
;
c
;
z
)
notation for Gaussβ hypergeometric function
,
F
q
p
β‘
(
a
1
,
β¦
,
a
p
;
b
1
,
β¦
,
b
q
;
z
)
or
F
q
p
β‘
(
a
1
,
β¦
,
a
p
b
1
,
β¦
,
b
q
;
z
)
: alternatively
F
q
p
β‘
(
π
;
π
;
z
)
or
F
q
p
β‘
(
π
π
;
z
)
generalized hypergeometric function
,
z
: complex variable
,
a
,
a
1
,
β¦
,
a
p
: real or complex parameters
and
b
,
b
1
,
β¦
,
b
q
: real or complex parameters
Referenced by:
Β§16.12
Permalink:
http://dlmf.nist.gov/16.12.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
Β§16.12
and
Ch.16
More generally,
16.12.3
(
F
1
2
β‘
(
a
,
b
c
;
z
)
)
2
=
β
k
=
0
β
(
2
β’
a
)
k
β’
(
2
β’
b
)
k
β’
(
c
β
1
2
)
k
(
c
)
k
β’
(
2
β’
c
β
1
)
k
β’
k
!
β’
F
3
4
β‘
(
β
1
2
β’
k
,
1
2
β’
(
1
β
k
)
,
a
+
b
β
c
+
1
2
,
1
2
a
+
1
2
,
b
+
1
2
,
3
2
β
k
β
c
;
1
)
β’
z
k
,
|
z
|
<
1
.
β
Symbols:
F
1
2
β‘
(
a
,
b
;
c
;
z
)
:
=
F
β‘
(
a
,
b
;
c
;
z
)
notation for Gaussβ hypergeometric function
,
F
q
p
β‘
(
a
1
,
β¦
,
a
p
;
b
1
,
β¦
,
b
q
;
z
)
or
F
q
p
β‘
(
a
1
,
β¦
,
a
p
b
1
,
β¦
,
b
q
;
z
)
: alternatively
F
q
p
β‘
(
π
;
π
;
z
)
or
F
q
p
β‘
(
π
π
;
z
)
generalized hypergeometric function
,
(
a
)
n
: Pochhammerβs symbol (or shifted factorial)
,
!
: factorial (as in
n
!
)
,
z
: complex variable
,
a
,
a
1
,
β¦
,
a
p
: real or complex parameters
and
b
,
b
1
,
β¦
,
b
q
: real or complex parameters
Referenced by:
Β§16.12
Permalink:
http://dlmf.nist.gov/16.12.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
Β§16.12
and
Ch.16
For further identities see
Goursat (
1883
)
and
ErdΓ©lyi
et al.
(
1953a
, Β§4.3)
.