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Formelsammlung Mathematik: Unendliche Reihen: Reihen zur Jacobischen Thetafunktion
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Unendliche Reihen
Es sei
ϑ
3
(
q
)
=
∑
n
∈
Z
q
n
2
{\displaystyle \vartheta _{3}(q)=\sum _{n\in \mathbb {Z} }q^{n^{2}}}
und
a
m
:=
Γ
(
3
4
)
π
1
4
ϑ
3
(
e
−
m
π
)
{\displaystyle a_{m}:={\frac {\Gamma \left({\frac {3}{4}}\right)}{\pi ^{\frac {1}{4}}}}\,\vartheta _{3}\left(e^{-m\pi }\right)}
. Vermutlich wird
a
m
∀
m
∈
Z
>
0
{\displaystyle a_{m}\quad \forall m\in \mathbb {Z} ^{>0}}
algebraisch sein.
m
{\displaystyle m\,}
Minimalpolynom
m
{\displaystyle m\,}
von
a
m
{\displaystyle a_{m}\,}
deg
m
{\displaystyle {\text{deg}}\,m\,}
a
m
{\displaystyle a_{m}\,}
ausgedrückt durch Radikale
1
{\displaystyle 1\,}
x
−
1
{\displaystyle x-1\,}
1
{\displaystyle 1\,}
1
{\displaystyle 1\,}
2
{\displaystyle 2\,}
8
x
4
−
8
x
2
+
1
{\displaystyle 8x^{4}-8x^{2}+1\,}
4
{\displaystyle 4\,}
2
+
2
2
{\displaystyle {\frac {\sqrt {{\sqrt {2}}+2}}{2}}}
3
{\displaystyle 3\,}
27
x
8
−
18
x
4
−
1
{\displaystyle 27x^{8}-18x^{4}-1\,}
8
{\displaystyle 8\,}
2
+
3
3
3
4
{\displaystyle {\sqrt[{4}]{\frac {2+{\sqrt {3}}}{3{\sqrt {3}}}}}}
4
{\displaystyle 4\,}
32
x
4
−
64
x
3
+
48
x
2
−
16
x
+
1
{\displaystyle 32x^{4}-64x^{3}+48x^{2}-16x+1\,}
4
{\displaystyle 4\,}
2
+
2
4
3
4
{\displaystyle {\frac {2+{\sqrt[{4}]{2}}^{3}}{4}}}
5
{\displaystyle 5\,}
25
x
4
−
20
x
2
−
1
{\displaystyle 25x^{4}-20x^{2}-1\,}
4
{\displaystyle 4\,}
5
+
2
5
{\displaystyle {\sqrt {\frac {{\sqrt {5}}+2}{5}}}}
6
{\displaystyle 6\,}
−
4
+
3
2
+
3
4
5
+
2
3
−
3
4
3
+
2
2
3
4
3
3
2
3
8
3
(
2
−
1
)
(
3
−
1
)
6
{\displaystyle {\frac {\sqrt[{3}]{-4+3{\sqrt {2}}+{\sqrt[{4}]{3}}^{5}+2{\sqrt {3}}-{\sqrt[{4}]{3}}^{3}+2{\sqrt {2}}{\sqrt[{4}]{3}}^{3}}}{2\,{\sqrt[{8}]{3}}^{3}\,{\sqrt[{6}]{({\sqrt {2}}-1)({\sqrt {3}}-1)}}}}}
7
{\displaystyle 7\,}
823543
x
16
−
470596
x
12
−
72030
x
8
−
10388
x
4
−
1
{\displaystyle 823543x^{16}-470596x^{12}-72030x^{8}-10388x^{4}-1\,}
16
{\displaystyle 16\,}
7
+
4
7
+
2
35
+
16
7
4
7
{\displaystyle {\frac {\sqrt[{4}]{7+4{\sqrt {7}}+2{\sqrt {35+16{\sqrt {7}}}}}}{\sqrt {7}}}}
8
{\displaystyle 8\,}
268435456
x
16
−
268435456
x
14
+
.
.
.
−
84608
x
2
+
1
{\displaystyle 268435456x^{16}-268435456x^{14}+...-84608x^{2}+1\,}
16
{\displaystyle 16\,}
9
{\displaystyle 9\,}
243
x
6
−
486
x
5
+
405
x
4
−
216
x
3
+
81
x
2
−
18
x
−
1
{\displaystyle 243x^{6}-486x^{5}+405x^{4}-216x^{3}+81x^{2}-18x-1\,}
6
{\displaystyle 6\,}
1
+
2
+
2
3
3
3
{\displaystyle {\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}}
10
{\displaystyle 10\,}
1600000000
x
16
−
2560000000
x
14
+
.
.
.
−
1958080
x
2
+
1
{\displaystyle 1600000000x^{16}-2560000000x^{14}+...-1958080x^{2}+1\,}
16
{\displaystyle 16\,}
12
{\displaystyle 12\,}
32
{\displaystyle 32\,}
ϑ
3
(
e
−
2
π
)
=
Γ
(
9
8
)
Γ
(
5
4
)
Γ
(
1
4
)
2
4
π
{\displaystyle \vartheta _{3}\left(e^{-{\sqrt {2}}\pi }\right)={\frac {\Gamma \left({\frac {9}{8}}\right)}{\Gamma \left({\frac {5}{4}}\right)}}\,{\sqrt {\frac {\Gamma \left({\frac {1}{4}}\right)}{{\sqrt[{4}]{2}}\,\pi }}}}
ϑ
3
(
e
−
3
π
)
=
2
3
27
8
Γ
(
1
3
)
Γ
(
2
3
)
{\displaystyle \vartheta _{3}\left(e^{-{\sqrt {3}}\pi }\right)={\frac {\sqrt[{3}]{2}}{\sqrt[{8}]{27}}}\,{\frac {\sqrt {\Gamma \left({\frac {1}{3}}\right)}}{\Gamma \left({\frac {2}{3}}\right)}}}
und 
. Vermutlich wird 
algebraisch sein.
von 
ausgedrückt durch Radikale
![{\displaystyle {\sqrt[{4}]{\frac {2+{\sqrt {3}}}{3{\sqrt {3}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1012d661555ee8531e3ec7519d8bf750905821)
![{\displaystyle {\frac {2+{\sqrt[{4}]{2}}^{3}}{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd92df03a362ebf0c1771236c867b0b00a144c5)
![{\displaystyle {\frac {\sqrt[{3}]{-4+3{\sqrt {2}}+{\sqrt[{4}]{3}}^{5}+2{\sqrt {3}}-{\sqrt[{4}]{3}}^{3}+2{\sqrt {2}}{\sqrt[{4}]{3}}^{3}}}{2\,{\sqrt[{8}]{3}}^{3}\,{\sqrt[{6}]{({\sqrt {2}}-1)({\sqrt {3}}-1)}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4d47d3af97e3c622eae9b6dfb9f5b12ba1866b)
![{\displaystyle {\frac {\sqrt[{4}]{7+4{\sqrt {7}}+2{\sqrt {35+16{\sqrt {7}}}}}}{\sqrt {7}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac502ab4ee6624c05918b18f184c2d4957fa8b9d)
![{\displaystyle {\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d5a2c1246c76e46b3b70296f56b7dc1623639a2)
![{\displaystyle \vartheta _{3}\left(e^{-{\sqrt {2}}\pi }\right)={\frac {\Gamma \left({\frac {9}{8}}\right)}{\Gamma \left({\frac {5}{4}}\right)}}\,{\sqrt {\frac {\Gamma \left({\frac {1}{4}}\right)}{{\sqrt[{4}]{2}}\,\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95a414feb5e965efb00ae8aa1efbc62b8b0fa67b)
![{\displaystyle \vartheta _{3}\left(e^{-{\sqrt {3}}\pi }\right)={\frac {\sqrt[{3}]{2}}{\sqrt[{8}]{27}}}\,{\frac {\sqrt {\Gamma \left({\frac {1}{3}}\right)}}{\Gamma \left({\frac {2}{3}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24a6dd303887e9884b9cd9ab0963888f76fe4313)
