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Statements
w
6
(
z
)
=
e
b
π
i
z
-
b
F
(
b
,
b
-
c
+
1
b
-
a
+
1
;
1
z
)
=
e
(
c
-
a
)
π
i
z
a
-
c
(
1
-
z
)
c
-
a
-
b
F
(
1
-
a
,
c
-
a
b
-
a
+
1
;
1
z
)
=
(
1
-
z
)
-
b
F
(
b
,
c
-
a
b
-
a
+
1
;
1
1
-
z
)
=
e
(
c
-
1
)
π
i
z
1
-
c
(
1
-
z
)
c
-
b
-
1
F
(
1
-
a
,
b
-
c
+
1
b
-
a
+
1
;
1
1
-
z
)
.
subscript
𝑤
6
𝑧
superscript
𝑒
𝑏
𝜋
imaginary-unit
superscript
𝑧
𝑏
Gauss-hypergeometric-F
𝑏
𝑏
𝑐
1
𝑏
𝑎
1
1
𝑧
superscript
𝑒
𝑐
𝑎
𝜋
imaginary-unit
superscript
𝑧
𝑎
𝑐
superscript
1
𝑧
𝑐
𝑎
𝑏
Gauss-hypergeometric-F
1
𝑎
𝑐
𝑎
𝑏
𝑎
1
1
𝑧
superscript
1
𝑧
𝑏
Gauss-hypergeometric-F
𝑏
𝑐
𝑎
𝑏
𝑎
1
1
1
𝑧
superscript
𝑒
𝑐
1
𝜋
imaginary-unit
superscript
𝑧
1
𝑐
superscript
1
𝑧
𝑐
𝑏
1
Gauss-hypergeometric-F
1
𝑎
𝑏
𝑐
1
𝑏
𝑎
1
1
1
𝑧
{\displaystyle{\displaystyle w_{6}(z)=e^{b\pi\mathrm{i}}z^{-b}F\left({b,b-c+1%
\atop b-a+1};\frac{1}{z}\right)=e^{(c-a)\pi\mathrm{i}}z^{a-c}(1-z)^{c-a-b}\*F%
\left({1-a,c-a\atop b-a+1};\frac{1}{z}\right)=(1-z)^{-b}F\left({b,c-a\atop b-a%
+1};\frac{1}{1-z}\right)=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-b-1}\*F\left({1%
-a,b-c+1\atop b-a+1};\frac{1}{1-z}\right).}}
F
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
𝑎
{\displaystyle{\displaystyle a}}
b
𝑏
{\displaystyle{\displaystyle b}}
c
𝑐
{\displaystyle{\displaystyle c}}
w
m
(
z
)
subscript
𝑤
𝑚
𝑧
{\displaystyle{\displaystyle w_{m}(z)}}
