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Testing:
(
c
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2
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1
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+
(
2
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2
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=
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{\displaystyle \displaystyle {}(c-a){}_{2}F_{1}(a-1,b;c;z)+(2a-c-az+bz){}_{2}F_{1}(a,b;c;z)+a(z-1){}_{2}F_{1}(a+1,b;c;z)=0}
(
c
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2
F
1
(
a
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b
−
1
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c
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+
(
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2
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1
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1
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=
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{\displaystyle \displaystyle {}(c-b){}_{2}F_{1}(a,b-1;c;z)+(2b-c-bz+az){}_{2}F_{1}(a,b;c;z)+b(z-1){}_{2}F_{1}(a,b+1;c;z)=0}
c
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2
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1
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[
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2
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2
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(
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2
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1
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=
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{\displaystyle \displaystyle {}c(c-1)(z-1){}_{2}F_{1}(a,b;c-1;z)+c[c-1-(2c-a-b-1)z]{}_{2}F_{1}(a,b;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}
[
c
−
2
a
−
(
b
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a
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z
]
2
F
1
(
a
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a
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1
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2
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−
(
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2
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1
(
a
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1
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b
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c
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{\displaystyle \displaystyle {}[c-2a-(b-a)z]{}_{2}F_{1}(a,b;c;z)+a(1-z){}_{2}F_{1}(a+1,b;c;z)-(c-a){}_{2}F_{1}(a-1,b;c;z)=0}
(
b
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2
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1
(
a
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b
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c
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+
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2
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2
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(
a
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1
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c
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=
0
{\displaystyle \displaystyle {}(b-a){}_{2}F_{1}(a,b;c;z)+a{}_{2}F_{1}(a+1,b;c;z)-b{}_{2}F_{1}(a,b+1;c;z)=0}
(
c
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2
F
1
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a
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+
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−
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2
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1
(
a
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1
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c
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{\displaystyle \displaystyle {}(c-a-b){}_{2}F_{1}(a,b;c;z)+a(1-z){}_{2}F_{1}(a+1,b;c;z)-(c-b){}_{2}F_{1}(a,b-1;c;z)=0}
c
[
a
−
(
c
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z
]
2
F
1
(
a
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c
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1
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2
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1
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a
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{\displaystyle \displaystyle {}c[a-(c-b)z]{}_{2}F_{1}(a,b;c;z)-ac(1-z){}_{2}F_{1}(a+1,b;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}
(
c
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2
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(
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2
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c
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2
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1
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a
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b
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c
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{\displaystyle \displaystyle {}(c-a-1){}_{2}F_{1}(a,b;c;z)+a{}_{2}F_{1}(a+1,b;c;z)-(c-1){}_{2}F_{1}(a,b;c-1;z)=0}
(
c
−
a
−
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2
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1
(
a
,
b
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−
(
c
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1
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a
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{\displaystyle \displaystyle {}(c-a-b){}_{2}F_{1}(a,b;c;z)-(c-a){}_{2}F_{1}(a-1,b;c;z)+b(1-z){}_{2}F_{1}(a,b+1;c;z)=0}
(
b
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(
1
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2
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b
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a
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1
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{\displaystyle \displaystyle {}(b-a)(1-z){}_{2}F_{1}(a,b;c;z)-(c-a){}_{2}F_{1}(a-1,b;c;z)+(c-b){}_{2}F_{1}(a,b-1;c;z)=0}
c
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2
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1
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b
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−
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2
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{\displaystyle \displaystyle {}c(1-z){}_{2}F_{1}(a,b;c;z)-c{}_{2}F_{1}(a-1,b;c;z)+(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}
[
a
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1
−
(
c
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1
)
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]
2
F
1
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1
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2
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1
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a
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b
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c
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{\displaystyle \displaystyle {}[a-1-(c-b-1)z]{}_{2}F_{1}(a,b;c;z)+(c-a){}_{2}F_{1}(a-1,b;c;z)-(c-1)(1-z){}_{2}F_{1}(a,b;c-1;z)=0}
[
c
−
2
b
+
(
b
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a
)
z
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2
F
1
(
a
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b
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1
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a
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{\displaystyle \displaystyle {}[c-2b+(b-a)z]{}_{2}F_{1}(a,b;c;z)+b(1-z){}_{2}F_{1}(a,b+1;c;z)-(c-b){}_{2}F_{1}(a,b-1;c;z)=0}
c
[
b
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2
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1
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{\displaystyle \displaystyle {}c[b-(c-a)z]{}_{2}F_{1}(a,b;c;z)-bc(1-z){}_{2}F_{1}(a,b+1;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}
(
c
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2
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(
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1
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{\displaystyle \displaystyle {}(c-b-1){}_{2}F_{1}(a,b;c;z)+b{}_{2}F_{1}(a,b+1;c;z)-(c-1){}_{2}F_{1}(a,b;c-1;z)=0}
c
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2
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2
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a
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1
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{\displaystyle \displaystyle {}c(1-z){}_{2}F_{1}(a,b;c;z)-c{}_{2}F_{1}(a,b-1;c;z)+(c-a)z{}_{2}F_{1}(a,b;c+1;z)=0}
[
b
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1
−
(
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a
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2
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1
(
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1
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2
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1
(
a
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b
;
c
−
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0
{\displaystyle \displaystyle {}[b-1-(c-a-1)z]{}_{2}F_{1}(a,b;c;z)+(c-b){}_{2}F_{1}(a,b-1;c;z)-(c-1)(1-z){}_{2}F_{1}(a,b;c-1;z)=0}
c
[
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−
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2
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]
2
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1
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2
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1
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−
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c
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1
−
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2
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1
(
a
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b
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c
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1
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=
0
{\displaystyle \displaystyle {}c[c-1-(2c-a-b-1)z]{}_{2}F_{1}(a,b;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)-c(c-1)(1-z){}_{2}F_{1}(a,b;c-1;z)=0}
[
c
−
2
a
−
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b
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a
)
z
]
F
+
a
(
1
−
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)
F
a
+
−
(
c
−
a
)
F
a
−
=
0
{\displaystyle \displaystyle {}[c-2a-(b-a)z]F+a(1-z)F^{a+}-(c-a)F^{a-}=0}
(
b
−
a
)
F
+
a
F
a
+
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F
b
+
=
0
{\displaystyle \displaystyle {}(b-a)F+aF^{a+}-bF^{b+}=0}
(
c
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a
−
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F
+
a
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1
−
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F
a
+
−
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c
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F
b
−
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0
{\displaystyle \displaystyle {}(c-a-b)F+a(1-z)F^{a+}-(c-b)F^{b-}=0}
c
[
a
−
(
c
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)
z
]
F
−
a
c
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1
−
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)
F
a
+
+
(
c
−
a
)
(
c
−
b
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z
F
c
+
=
0
{\displaystyle \displaystyle {}c[a-(c-b)z]F-ac(1-z)F^{a+}+(c-a)(c-b)zF^{c+}=0}
(
c
−
a
−
1
)
F
+
a
F
a
+
−
(
c
−
1
)
F
c
−
=
0
{\displaystyle \displaystyle {}(c-a-1)F+aF^{a+}-(c-1)F^{c-}=0}
(
c
−
a
−
b
)
F
−
(
c
−
a
)
F
a
−
+
b
(
1
−
z
)
F
b
+
=
0
{\displaystyle \displaystyle {}(c-a-b)F-(c-a)F^{a-}+b(1-z)F^{b+}=0}
(
b
−
a
)
(
1
−
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)
F
−
(
c
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a
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F
a
−
+
(
c
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b
)
F
b
−
=
0
{\displaystyle \displaystyle {}(b-a)(1-z)F-(c-a)F^{a-}+(c-b)F^{b-}=0}
c
(
1
−
z
)
F
−
c
F
a
−
+
(
c
−
b
)
z
F
c
+
=
0
{\displaystyle \displaystyle {}c(1-z)F-cF^{a-}+(c-b)zF^{c+}=0}
[
a
−
1
−
(
c
−
b
−
1
)
z
]
F
+
(
c
−
a
)
F
a
−
−
(
c
−
1
)
(
1
−
z
)
F
c
−
=
0
{\displaystyle \displaystyle {}[a-1-(c-b-1)z]F+(c-a)F^{a-}-(c-1)(1-z)F^{c-}=0}
[
c
−
2
b
+
(
b
−
a
)
z
]
F
+
b
(
1
−
z
)
F
b
+
−
(
c
−
b
)
F
b
−
=
0
{\displaystyle \displaystyle {}[c-2b+(b-a)z]F+b(1-z)F^{b+}-(c-b)F^{b-}=0}
c
[
b
−
(
c
−
a
)
z
]
F
−
b
c
(
1
−
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)
F
b
+
+
(
c
−
a
)
(
c
−
b
)
z
F
c
+
=
0
{\displaystyle \displaystyle {}c[b-(c-a)z]F-bc(1-z)F^{b+}+(c-a)(c-b)zF^{c+}=0}
(
c
−
b
−
1
)
F
+
b
F
b
+
−
(
c
−
1
)
F
c
−
=
0
{\displaystyle \displaystyle {}(c-b-1)F+bF^{b+}-(c-1)F^{c-}=0}
c
(
1
−
z
)
F
−
c
F
b
−
+
(
c
−
a
)
z
F
c
+
=
0
{\displaystyle \displaystyle {}c(1-z)F-cF^{b-}+(c-a)zF^{c+}=0}
[
b
−
1
−
(
c
−
a
−
1
)
z
]
F
+
(
c
−
b
)
F
b
−
−
(
c
−
1
)
(
1
−
z
)
F
c
−
=
0
{\displaystyle \displaystyle {}[b-1-(c-a-1)z]F+(c-b)F^{b-}-(c-1)(1-z)F^{c-}=0}
c
[
c
−
1
−
(
2
c
−
a
−
b
−
1
)
z
]
F
+
(
c
−
a
)
(
c
−
b
)
z
F
c
+
−
c
(
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1
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F
c
−
=
0
{\displaystyle \displaystyle {}c[c-1-(2c-a-b-1)z]F+(c-a)(c-b)zF^{c+}-c(c-1)(1-z)F^{c-}=0}
Abramowitz, Milton
;
Stegun, Irene Ann
, eds. (1983) [June 1964].
"Chapter 15"
.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 558.
ISBN
978-0-486-61272-0
.
LCCN
64-60036
.
MR
0167642
.
LCCN
65-12253
.
![{\displaystyle \displaystyle {}c(c-1)(z-1){}_{2}F_{1}(a,b;c-1;z)+c[c-1-(2c-a-b-1)z]{}_{2}F_{1}(a,b;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e18c7ab25f96f7baae29ab2087523445bbab46b2)
![{\displaystyle \displaystyle {}[c-2a-(b-a)z]{}_{2}F_{1}(a,b;c;z)+a(1-z){}_{2}F_{1}(a+1,b;c;z)-(c-a){}_{2}F_{1}(a-1,b;c;z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d069f47d3da3a42e8fb45555059a5815db5d6b)
![{\displaystyle \displaystyle {}c[a-(c-b)z]{}_{2}F_{1}(a,b;c;z)-ac(1-z){}_{2}F_{1}(a+1,b;c;z)+(c-a)(c-b)z{}_{2}F_{1}(a,b;c+1;z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b742e9ae5beac17d2c50dc1cffffde299a1d3bea)
![{\displaystyle \displaystyle {}[a-1-(c-b-1)z]{}_{2}F_{1}(a,b;c;z)+(c-a){}_{2}F_{1}(a-1,b;c;z)-(c-1)(1-z){}_{2}F_{1}(a,b;c-1;z)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd00d15a42b1d9567f2d286a34b51e175f44f26)
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