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Statements
F
(
a
,
b
-
λ
c
+
λ
;
-
z
)
=
2
b
-
c
+
(
1
/
2
)
(
z
+
1
2
z
)
λ
(
λ
a
/
2
U
(
a
-
1
2
,
-
α
λ
)
(
(
1
+
z
)
c
-
a
-
b
z
1
-
c
(
α
z
-
1
)
1
-
a
+
O
(
λ
-
1
)
)
+
λ
(
a
-
1
)
/
2
α
U
(
a
-
3
2
,
-
α
λ
)
(
(
1
+
z
)
c
-
a
-
b
z
1
-
c
(
α
z
-
1
)
1
-
a
-
2
c
-
b
-
(
1
/
2
)
(
α
z
-
1
)
a
+
O
(
λ
-
1
)
)
)
,
Gauss-hypergeometric-F
𝑎
𝑏
𝜆
𝑐
𝜆
𝑧
superscript
2
𝑏
𝑐
1
2
superscript
𝑧
1
2
𝑧
𝜆
superscript
𝜆
𝑎
2
parabolic-U
𝑎
1
2
𝛼
𝜆
superscript
1
𝑧
𝑐
𝑎
𝑏
superscript
𝑧
1
𝑐
superscript
𝛼
𝑧
1
1
𝑎
Big-O
superscript
𝜆
1
superscript
𝜆
𝑎
1
2
𝛼
parabolic-U
𝑎
3
2
𝛼
𝜆
superscript
1
𝑧
𝑐
𝑎
𝑏
superscript
𝑧
1
𝑐
superscript
𝛼
𝑧
1
1
𝑎
superscript
2
𝑐
𝑏
1
2
superscript
𝛼
𝑧
1
𝑎
Big-O
superscript
𝜆
1
{\displaystyle{\displaystyle F\left({a,b-\lambda\atop c+\lambda};-z\right)=2^{%
b-c+(1/2)}\left(\frac{z+1}{2\sqrt{z}}\right)^{\lambda}\left(\lambda^{a/2}U%
\left(a-\tfrac{1}{2},-\alpha\sqrt{\lambda}\right)\left((1+z)^{c-a-b}z^{1-c}%
\left(\frac{\alpha}{z-1}\right)^{1-a}+O(\lambda^{-1})\right)+\frac{\lambda^{%
\ifrac{(a-1)}{2}}}{\alpha}U\left(a-\tfrac{3}{2},-\alpha\sqrt{\lambda}\right)%
\left((1+z)^{c-a-b}z^{1-c}\left(\frac{\alpha}{z-1}\right)^{1-a}-2^{c-b-(\ifrac%
{1}{2})}\left(\frac{\alpha}{z-1}\right)^{a}+O(\lambda^{-1})\right)\right),}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\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)}}
U
(
a
,
z
)
parabolic-U
𝑎
𝑧
{\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
𝑎
{\displaystyle{\displaystyle a}}
b
𝑏
{\displaystyle{\displaystyle b}}
c
𝑐
{\displaystyle{\displaystyle c}}
α
𝛼
{\displaystyle{\displaystyle\alpha}}
Identifiers
