# Formelsammlung Mathematik: Unendliche Reihen: Reihen zur Jacobischen Thetafunktion

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Es sei ${\displaystyle \vartheta _{3}(q)=\sum _{n\in \mathbb {Z} }q^{n^{2}}}$ und ${\displaystyle a_{m}:={\frac {\Gamma \left({\frac {3}{4}}\right)}{\pi ^{\frac {1}{4}}}}\,\vartheta _{3}\left(e^{-m\pi }\right)}$. Vermutlich wird ${\displaystyle a_{m}\quad \forall m\in \mathbb {Z} ^{>0}}$ algebraisch sein.
${\displaystyle m\,}$
Minimalpolynom ${\displaystyle m\,}$ von ${\displaystyle a_{m}\,}$ ${\displaystyle {\text{deg}}\,m\,}$ ${\displaystyle a_{m}\,}$ ausgedrückt durch Radikale
${\displaystyle 1\,}$ ${\displaystyle x-1\,}$ ${\displaystyle 1\,}$ ${\displaystyle 1\,}$
${\displaystyle 2\,}$ ${\displaystyle 8x^{4}-8x^{2}+1\,}$ ${\displaystyle 4\,}$ ${\displaystyle {\frac {\sqrt {{\sqrt {2}}+2}}{2}}}$
${\displaystyle 3\,}$ ${\displaystyle 27x^{8}-18x^{4}-1\,}$ ${\displaystyle 8\,}$ ${\displaystyle {\sqrt[{4}]{\frac {2+{\sqrt {3}}}{3{\sqrt {3}}}}}}$
${\displaystyle 4\,}$ ${\displaystyle 32x^{4}-64x^{3}+48x^{2}-16x+1\,}$ ${\displaystyle 4\,}$ ${\displaystyle {\frac {2+{\sqrt[{4}]{2}}^{3}}{4}}}$
${\displaystyle 5\,}$ ${\displaystyle 25x^{4}-20x^{2}-1\,}$ ${\displaystyle 4\,}$ ${\displaystyle {\sqrt {\frac {{\sqrt {5}}+2}{5}}}}$
${\displaystyle 6\,}$ ${\displaystyle {\frac {\sqrt[{3}]{-4+3{\sqrt {2}}+{\sqrt[{4}]{3}}^{5}+2{\sqrt {3}}-{\sqrt[{4}]{3}}^{3}+2{\sqrt {2}}{\sqrt[{4}]{3}}^{3}}}{2\,{\sqrt[{8}]{3}}^{3}\,{\sqrt[{6}]{({\sqrt {2}}-1)({\sqrt {3}}-1)}}}}}$
${\displaystyle 7\,}$ ${\displaystyle 823543x^{16}-470596x^{12}-72030x^{8}-10388x^{4}-1\,}$ ${\displaystyle 16\,}$ ${\displaystyle {\frac {\sqrt[{4}]{7+4{\sqrt {7}}+2{\sqrt {35+16{\sqrt {7}}}}}}{\sqrt {7}}}}$
${\displaystyle 8\,}$ ${\displaystyle 268435456x^{16}-268435456x^{14}+...-84608x^{2}+1\,}$ ${\displaystyle 16\,}$
${\displaystyle 9\,}$ ${\displaystyle 243x^{6}-486x^{5}+405x^{4}-216x^{3}+81x^{2}-18x-1\,}$ ${\displaystyle 6\,}$ ${\displaystyle {\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}}$
${\displaystyle 10\,}$ ${\displaystyle 1600000000x^{16}-2560000000x^{14}+...-1958080x^{2}+1\,}$ ${\displaystyle 16\,}$
${\displaystyle 12\,}$ ${\displaystyle 32\,}$
${\displaystyle \vartheta _{3}\left(e^{-{\sqrt {2}}\pi }\right)={\frac {\Gamma \left({\frac {9}{8}}\right)}{\Gamma \left({\frac {5}{4}}\right)}}\,{\sqrt {\frac {\Gamma \left({\frac {1}{4}}\right)}{{\sqrt[{4}]{2}}\,\pi }}}}$

${\displaystyle \vartheta _{3}\left(e^{-{\sqrt {3}}\pi }\right)={\frac {\sqrt[{3}]{2}}{\sqrt[{8}]{27}}}\,{\frac {\sqrt {\Gamma \left({\frac {1}{3}}\right)}}{\Gamma \left({\frac {2}{3}}\right)}}}$