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Statements
ϕ
1
2
(
a
,
b
c
;
q
,
z
)
=
(
b
,
c
/
a
,
a
z
,
q
/
(
a
z
)
;
q
)
∞
(
c
,
b
/
a
,
z
,
q
/
z
;
q
)
∞
ϕ
1
2
(
a
,
a
q
/
c
a
q
/
b
;
q
,
c
q
/
(
a
b
z
)
)
+
(
a
,
c
/
b
,
b
z
,
q
/
(
b
z
)
;
q
)
∞
(
c
,
a
/
b
,
z
,
q
/
z
;
q
)
∞
ϕ
1
2
(
b
,
b
q
/
c
b
q
/
a
;
q
,
c
q
/
(
a
b
z
)
)
,
q-hypergeometric-rphis
2
1
𝑎
𝑏
𝑐
𝑞
𝑧
q-multiple-Pochhammer
𝑏
𝑐
𝑎
𝑎
𝑧
𝑞
𝑎
𝑧
𝑞
q-multiple-Pochhammer
𝑐
𝑏
𝑎
𝑧
𝑞
𝑧
𝑞
q-hypergeometric-rphis
2
1
𝑎
𝑎
𝑞
𝑐
𝑎
𝑞
𝑏
𝑞
𝑐
𝑞
𝑎
𝑏
𝑧
q-multiple-Pochhammer
𝑎
𝑐
𝑏
𝑏
𝑧
𝑞
𝑏
𝑧
𝑞
q-multiple-Pochhammer
𝑐
𝑎
𝑏
𝑧
𝑞
𝑧
𝑞
q-hypergeometric-rphis
2
1
𝑏
𝑏
𝑞
𝑐
𝑏
𝑞
𝑎
𝑞
𝑐
𝑞
𝑎
𝑏
𝑧
{\displaystyle{\displaystyle{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=%
\frac{\left(b,c/a,az,q/(az);q\right)_{\infty}}{\left(c,b/a,z,q/z;q\right)_{%
\infty}}{{}_{2}\phi_{1}}\left({a,aq/c\atop aq/b};q,cq/(abz)\right)+\frac{\left%
(a,c/b,bz,q/(bz);q\right)_{\infty}}{\left(c,a/b,z,q/z;q\right)_{\infty}}{{}_{2%
}\phi_{1}}\left({b,bq/c\atop bq/a};q,cq/(abz)\right),}}
|
a
b
z
|
<
|
c
q
|
𝑎
𝑏
𝑧
𝑐
𝑞
{\displaystyle{\displaystyle|abz|<|cq|}}
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
q-hypergeometric-rphis
𝑟
1
𝑠
subscript
𝑎
0
…
subscript
𝑎
𝑟
subscript
𝑏
1
…
subscript
𝑏
𝑠
𝑞
𝑧
{\displaystyle{\displaystyle{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},%
\dots,a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)}}
(
a
1
,
a
2
,
…
,
a
r
;
q
)
n
q-multiple-Pochhammer
subscript
𝑎
1
subscript
𝑎
2
…
subscript
𝑎
𝑟
𝑞
𝑛
{\displaystyle{\displaystyle\left(\NVar{a_{1},a_{2},\dots,a_{r}};\NVar{q}%
\right)_{\NVar{n}}}}
q
𝑞
{\displaystyle{\displaystyle q}}
z
𝑧
{\displaystyle{\displaystyle z}}
Identifiers
