利用者:HOTUMA/テータ関数の記号

ヤコービの記号

${\begin{aligned}\Theta (u)&=\vartheta _{4}(u/2K)\\\Theta _{1}(u)&=\vartheta _{3}(u/2K)\\\mathrm {H} (u)&=\vartheta _{1}(u/2K)\\\mathrm {H} _{1}(u)&=\vartheta _{2}(u/2K)\\\end{aligned}}$

ネビル(Neville)の記号

${\begin{aligned}\vartheta _{s}(u)&={\frac {\vartheta _{1}(u/2K)}{\vartheta _{1}'(0)}}\\\vartheta _{c}(u)&={\frac {\vartheta _{1}(u/2K+1/2)}{\vartheta _{1}(1/2)}}\\\vartheta _{d}(u)&={\frac {\vartheta _{3}(u/2K+1/2)}{\vartheta _{3}(1/2)}}\\\vartheta _{n}(u)&={\frac {\vartheta _{3}(u/2K)}{\vartheta _{3}(0)}}\\\end{aligned}}$

Abramowitz and Stegun: Handbook of Mathematical Functions, Tenth Printing, December 1972

q級数の記号

$Q_{0}=Q_{0}(q)=\prod _{n=1}^{\infty }(1-q^{2n})$

$Q_{1}=Q_{1}(q)=\prod _{n=1}^{\infty }(1+q^{2n})$

$Q_{2}=Q_{2}(q)=\prod _{n=1}^{\infty }(1+q^{2n-1})$

$Q_{3}=Q_{3}(q)=\prod _{n=1}^{\infty }(1-q^{2n-1})$

$Q_{0}(q)Q_{1}(q)=Q_{0}(q^{2})$

$Q_{2}(q)Q_{3}(q)=Q_{3}(q^{2})$

$Q_{0}(q)Q_{3}(q)=Q_{0}(q^{1/2})$

$Q_{1}(q)Q_{2}(q)=Q_{1}(q^{1/2})$

q-Pochhammer symbol

$(a)_{k}=(a;q)_{k}=\prod _{n=0}^{k-1}(1-aq^{k})$

$(q)_{\infty }=\prod _{n=1}^{\infty }(1-q^{k})$

$(a,b,\cdots ,c;q)_{k}=(a;q)_{k}(b;q)_{k}\cdots (c;q)_{k}$

$\left(a,b,\cdots ,c;d,e,\cdots ,f;q\right)_{k}={\frac {(a;q)_{k}(b;q)_{k}\cdots (c;q)_{k}}{(d;q)_{k}(e;q)_{k}\cdots (f;q)_{k}}}$

In a different convention,

$(a)=\prod _{n=1}^{\infty }(1-q^{an})$