Formelsammlung Mathematik: Unendliche Reihen: Reihen zur Jacobischen Thetafunktion

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Es sei $\vartheta _{3}(q)=\sum _{n\in \mathbb {Z} }q^{n^{2}}$ und $a_{m}:={\frac {\Gamma \left({\frac {3}{4}}\right)}{\pi ^{\frac {1}{4}}}}\,\vartheta _{3}\left(e^{-m\pi }\right)$. Vermutlich wird $a_{m}\quad \forall m\in \mathbb {Z} ^{>0}$ algebraisch sein.

$m\,$	Minimalpolynom $m\,$ von $a_{m}\,$	${\text{deg}}\,m\,$	$a_{m}\,$ ausgedrückt durch Radikale
$1\,$	$x-1\,$	$1\,$	$1\,$
$2\,$	$8x^{4}-8x^{2}+1\,$	$4\,$	${\frac {\sqrt {{\sqrt {2}}+2}}{2}}$
$3\,$	$27x^{8}-18x^{4}-1\,$	$8\,$	${\sqrt[{4}]{\frac {2+{\sqrt {3}}}{3{\sqrt {3}}}}}$
$4\,$	$32x^{4}-64x^{3}+48x^{2}-16x+1\,$	$4\,$	${\frac {2+{\sqrt[{4}]{2}}^{3}}{4}}$
$5\,$	$25x^{4}-20x^{2}-1\,$	$4\,$	${\sqrt {\frac {{\sqrt {5}}+2}{5}}}$
$6\,$			${\frac {\sqrt[{3}]{-4+3{\sqrt {2}}+{\sqrt[{4}]{3}}^{5}+2{\sqrt {3}}-{\sqrt[{4}]{3}}^{3}+2{\sqrt {2}}{\sqrt[{4}]{3}}^{3}}}{2\,{\sqrt[{8}]{3}}^{3}\,{\sqrt[{6}]{({\sqrt {2}}-1)({\sqrt {3}}-1)}}}}$
$7\,$	$823543x^{16}-470596x^{12}-72030x^{8}-10388x^{4}-1\,$	$16\,$	${\frac {\sqrt[{4}]{7+4{\sqrt {7}}+2{\sqrt {35+16{\sqrt {7}}}}}}{\sqrt {7}}}$
$8\,$	$268435456x^{16}-268435456x^{14}+...-84608x^{2}+1\,$	$16\,$
$9\,$	$243x^{6}-486x^{5}+405x^{4}-216x^{3}+81x^{2}-18x-1\,$	$6\,$	${\frac {1+{\sqrt[{3}]{2+2{\sqrt {3}}}}}{3}}$
$10\,$	$1600000000x^{16}-2560000000x^{14}+...-1958080x^{2}+1\,$	$16\,$
$12\,$		$32\,$

$\vartheta _{3}\left(e^{-{\sqrt {2}}\pi }\right)={\frac {\Gamma \left({\frac {9}{8}}\right)}{\Gamma \left({\frac {5}{4}}\right)}}\,{\sqrt {\frac {\Gamma \left({\frac {1}{4}}\right)}{{\sqrt[{4}]{2}}\,\pi }}}$

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