Formelsammlung Mathematik: Unendliche Reihen: Reihen zur Jacobischen Thetafunktion

Zurück zu Unendliche Reihen

Es sei ${\vartheta }_3(q)=\sum _{n\in \mathbb{Z}}{q}^{n^2}$ und $a_m:=\frac{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{{\pi }^{\frac{1}{4}}}{\vartheta }_3\left(e^{-m\pi }\right)$. Vermutlich wird $a_m\mathrm{\forall }m\in {\mathbb{Z}}^{>0}$ algebraisch sein.

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

${\vartheta }_3\left(e^{-\sqrt{2}\pi }\right)=\frac{\mathrm{\Gamma }\left(\frac{9}{8}\right)}{\mathrm{\Gamma }\left(\frac{5}{4}\right)}\sqrt{\frac{\mathrm{\Gamma }\left(\frac{1}{4}\right)}{\sqrt[4]{2}\pi }}$

${\vartheta }_3\left(e^{-\sqrt{3}\pi }\right)=\frac{\sqrt[3]{2}}{\sqrt[8]{27}}\frac{\sqrt{\mathrm{\Gamma }\left(\frac{1}{3}\right)}}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}$