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Statements
(
∏
ℓ
=
1
p
Γ
(
a
ℓ
)
/
∏
ℓ
=
1
q
Γ
(
b
ℓ
)
)
F
q
p
(
a
1
,
…
,
a
p
b
1
,
…
,
b
q
;
-
z
)
∼
E
p
,
q
(
z
e
-
π
i
)
+
E
p
,
q
(
z
e
π
i
)
,
asymptotic-expansion
superscript
subscript
product
ℓ
1
𝑝
Euler-Gamma
subscript
𝑎
ℓ
superscript
subscript
product
ℓ
1
𝑞
Euler-Gamma
subscript
𝑏
ℓ
Gauss-hypergeometric-pFq
𝑝
𝑞
subscript
𝑎
1
…
subscript
𝑎
𝑝
subscript
𝑏
1
…
subscript
𝑏
𝑞
𝑧
subscript
𝐸
𝑝
𝑞
𝑧
superscript
𝑒
𝜋
imaginary-unit
subscript
𝐸
𝑝
𝑞
𝑧
superscript
𝑒
𝜋
imaginary-unit
{\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{p}%
\Gamma\left(a_{\ell}\right)}{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}%
\right)}}\right){{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};%
-z\right)\sim E_{p,q}(ze^{-\pi\mathrm{i}})+E_{p,q}(ze^{\pi\mathrm{i}}),}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
F
q
p
(
a
1
,
…
,
a
p
;
b
1
,
…
,
b
q
;
z
)
Gauss-hypergeometric-pFq
𝑝
𝑞
subscript
𝑎
1
…
subscript
𝑎
𝑝
subscript
𝑏
1
…
subscript
𝑏
𝑞
𝑧
{\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots%
,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
∼
asymptotic-expansion
{\displaystyle{\displaystyle\sim}}
π
{\displaystyle{\displaystyle\pi}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
p
𝑝
{\displaystyle{\displaystyle p}}
q
𝑞
{\displaystyle{\displaystyle q}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
,
a
1
,
…
,
a
p
𝑎
subscript
𝑎
1
…
subscript
𝑎
𝑝
{\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
b
,
b
1
,
…
,
b
q
𝑏
subscript
𝑏
1
…
subscript
𝑏
𝑞
{\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
E
p
,
q
(
z
)
subscript
𝐸
𝑝
𝑞
𝑧
{\displaystyle{\displaystyle E_{p,q}(z)}}
e
k
,
m
subscript
𝑒
𝑘
𝑚
{\displaystyle{\displaystyle e_{k,m}}}
