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Statements
(
z
d
d
z
z
)
n
(
z
c
-
a
-
1
(
1
-
z
)
a
+
b
-
c
F
(
a
,
b
;
c
;
z
)
)
=
(
c
-
a
)
n
z
c
-
a
+
n
-
1
(
1
-
z
)
a
-
n
+
b
-
c
F
(
a
-
n
,
b
;
c
;
z
)
.
superscript
𝑧
derivative
𝑧
𝑧
𝑛
superscript
𝑧
𝑐
𝑎
1
superscript
1
𝑧
𝑎
𝑏
𝑐
Gauss-hypergeometric-F
𝑎
𝑏
𝑐
𝑧
Pochhammer
𝑐
𝑎
𝑛
superscript
𝑧
𝑐
𝑎
𝑛
1
superscript
1
𝑧
𝑎
𝑛
𝑏
𝑐
Gauss-hypergeometric-F
𝑎
𝑛
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}%
\left(z^{c-a-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-a\right)_{n}%
}z^{c-a+n-1}(1-z)^{a-n+b-c}\*F\left(a-n,b;c;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)}}
(
a
)
n
Pochhammer
𝑎
𝑛
{\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
d
f
d
x
derivative
𝑓
𝑥
{\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
n
𝑛
{\displaystyle{\displaystyle n}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
𝑎
{\displaystyle{\displaystyle a}}
b
𝑏
{\displaystyle{\displaystyle b}}
c
𝑐
{\displaystyle{\displaystyle c}}
