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NIST
17
q
-Hypergeometric and Related Functions
Properties
17.9
Further Transformations of
ϕ
r
r
+
1
Functions
17.11
Transformations of
q
-Appell Functions
§17.10
Transformations of
ψ
r
r
Functions
ⓘ
Keywords:
Bailey’s
ψ
2
2
transformations
,
bilateral
q
-hypergeometric function
Notes:
See
Gasper and Rahman (
2004
, pp. 147–150, 364–366)
.
Permalink:
http://dlmf.nist.gov/17.10
See also:
Annotations for
Ch.17
Bailey’s
ψ
2
2
Transformations
ⓘ
See also:
Annotations for
§17.10
and
Ch.17
17.10.1
ψ
2
2
(
a
,
b
c
,
d
;
q
,
z
)
=
(
a
z
,
d
/
a
,
c
/
b
,
d
q
/
(
a
b
z
)
;
q
)
∞
(
z
,
d
,
q
/
b
,
c
d
/
(
a
b
z
)
;
q
)
∞
ψ
2
2
(
a
,
a
b
z
/
d
a
z
,
c
;
q
,
d
a
)
,
ⓘ
Symbols:
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
,
q
: complex base
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/17.10.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17
17.10.2
ψ
2
2
(
a
,
b
c
,
d
;
q
,
z
)
=
(
a
z
,
b
z
,
c
q
/
(
a
b
z
)
,
d
q
/
(
a
b
z
)
;
q
)
∞
(
q
/
a
,
q
/
b
,
c
,
d
;
q
)
∞
×
ψ
2
2
(
a
b
z
/
c
,
a
b
z
/
d
a
z
,
b
z
;
q
,
c
d
a
b
z
)
.
ⓘ
Symbols:
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
,
q
: complex base
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/17.10.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17
Other Transformations
ⓘ
Keywords:
idem function
,
q
-hypergeometric function
See also:
Annotations for
§17.10
and
Ch.17
17.10.3
ψ
8
8
(
q
a
1
2
,
-
q
a
1
2
,
c
,
d
,
e
,
f
,
a
q
-
n
,
q
-
n
a
1
2
,
-
a
1
2
,
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
a
q
/
f
,
q
n
+
1
,
a
q
n
+
1
;
q
,
a
2
q
2
n
+
2
c
d
e
f
)
=
(
a
q
,
q
/
a
,
a
q
/
(
c
d
)
,
a
q
/
(
e
f
)
;
q
)
n
(
q
/
c
,
q
/
d
,
a
q
/
e
,
a
q
/
f
;
q
)
n
ψ
4
4
(
e
,
f
,
a
q
n
+
1
/
(
c
d
)
,
q
-
n
a
q
/
c
,
a
q
/
d
,
q
n
+
1
,
e
f
/
(
a
q
n
)
;
q
,
q
)
,
ⓘ
Symbols:
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
,
q
: complex base
and
n
: nonnegative integer
Permalink:
http://dlmf.nist.gov/17.10.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17
17.10.4
ψ
2
2
(
e
,
f
a
q
/
c
,
a
q
/
d
;
q
,
a
q
e
f
)
=
(
q
/
c
,
q
/
d
,
a
q
/
e
,
a
q
/
f
;
q
)
∞
(
a
q
,
q
/
a
,
a
q
/
(
c
d
)
,
a
q
/
(
e
f
)
;
q
)
∞
×
∑
n
=
-
∞
∞
(
1
-
a
q
2
n
)
(
c
,
d
,
e
,
f
;
q
)
n
(
1
-
a
)
(
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
a
q
/
f
;
q
)
n
(
q
a
3
c
d
e
f
)
n
q
n
2
.
ⓘ
Symbols:
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
,
q
: complex base
and
n
: nonnegative integer
Permalink:
http://dlmf.nist.gov/17.10.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17
17.10.5
(
a
q
/
b
,
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
q
/
(
a
b
)
,
q
/
(
a
c
)
,
q
/
(
a
d
)
,
q
/
(
a
e
)
;
q
)
∞
(
f
a
,
g
a
,
f
/
a
,
g
/
a
,
q
a
2
,
q
/
a
2
;
q
)
∞
×
ψ
8
8
(
q
a
,
-
q
a
,
b
a
,
c
a
,
d
a
,
e
a
,
f
a
,
g
a
a
,
-
a
,
a
q
/
b
,
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
a
q
/
f
,
a
q
/
g
;
q
,
q
2
b
c
d
e
f
g
)
=
(
q
,
q
/
(
b
f
)
,
q
/
(
c
f
)
,
q
/
(
d
f
)
,
q
/
(
e
f
)
,
q
f
/
b
,
q
f
/
c
,
q
f
/
d
,
q
f
/
e
;
q
)
∞
(
f
a
,
q
/
(
f
a
)
,
a
q
/
f
,
f
/
a
,
g
/
f
,
f
g
,
q
f
2
;
q
)
∞
×
ϕ
7
8
(
f
2
,
q
f
,
-
q
f
,
f
b
,
f
c
,
f
d
,
f
e
,
f
g
f
,
-
f
,
f
q
/
b
,
f
q
/
c
,
f
q
/
d
,
f
q
/
e
,
f
q
/
g
;
q
,
q
2
b
c
d
e
f
g
)
+
idem
(
f
;
g
)
.
ⓘ
Symbols:
idem
(
χ
1
;
χ
2
,
…
,
χ
n
)
:
idem
function
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
and
q
: complex base
Permalink:
http://dlmf.nist.gov/17.10.E5
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17
17.10.6
(
a
q
/
b
,
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
a
q
/
f
,
q
/
(
a
b
)
,
q
/
(
a
c
)
,
q
/
(
a
d
)
,
q
/
(
a
e
)
,
q
/
(
a
f
)
;
q
)
∞
(
a
g
,
a
h
,
a
k
,
g
/
a
,
h
/
a
,
k
/
a
,
q
a
2
,
q
/
a
2
;
q
)
∞
×
ψ
10
10
(
q
a
,
-
q
a
,
b
a
,
c
a
,
d
a
,
e
a
,
f
a
,
g
a
,
h
a
,
k
a
a
,
-
a
,
a
q
/
b
,
a
q
/
c
,
a
q
/
d
,
a
q
/
e
,
a
q
/
f
,
a
q
/
g
,
a
q
/
h
,
a
q
/
k
;
q
,
q
2
b
c
d
e
f
g
h
k
)
=
(
q
,
q
/
(
b
g
)
,
q
/
(
c
g
)
,
q
/
(
d
g
)
,
q
/
(
e
g
)
,
q
/
(
f
g
)
,
q
g
/
b
,
q
g
/
c
,
q
g
/
d
,
q
g
/
e
,
q
g
/
f
;
q
)
∞
(
g
h
,
g
k
,
h
/
g
,
a
g
,
q
/
(
a
g
)
,
g
/
a
,
a
q
/
g
,
q
g
2
;
q
)
∞
×
ϕ
9
10
(
g
2
,
q
g
,
-
q
g
,
g
b
,
g
c
,
g
d
,
g
e
,
g
f
,
g
h
,
g
k
g
,
-
g
,
q
g
/
b
,
q
g
/
c
,
q
g
/
d
,
q
g
/
e
,
q
g
/
f
,
q
g
/
h
,
q
g
/
k
;
q
,
q
2
b
c
d
e
f
g
h
k
)
+
idem
(
g
;
h
,
k
)
.
ⓘ
Symbols:
idem
(
χ
1
;
χ
2
,
…
,
χ
n
)
:
idem
function
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
ψ
s
r
(
a
1
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ψ
s
r
(
a
1
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: bilateral basic hypergeometric (or bilateral
q
-hypergeometric) function
,
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
: multiple
q
-Pochhammer symbol
,
k
: nonnegative integer
and
q
: complex base
Permalink:
http://dlmf.nist.gov/17.10.E6
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.10
,
§17.10
and
Ch.17