Ramanujan Theta Functions

Ramanujan's one-variable theta function is defined by

$\begin{displaymath} \varphi(x)\equiv \sum_{m=-\infty}^\infty x^{m^2}. \end{displaymath}$

(1)

It is equal to the function in the Jacobi Triple Product with

$\displaystyle \varphi(x)$	$\textstyle =$	$\displaystyle G(1)=\prod_{n=1}^\infty (1+x^{2n-1})^2(1-x^{2n})$
	$\textstyle =$	$\displaystyle \sum_{m=-\infty}^\infty x^{m^2} = 1+2\sum_{m=0}^\infty x^{m^2}.$	(2)

Special values include

$\displaystyle \varphi(e^{-\pi\sqrt{2}}\,)$	$\textstyle =$	$\displaystyle {\Gamma({\textstyle{9\over 8}})\over\Gamma({\textstyle{5\over 4}})}\sqrt{\Gamma({\textstyle{1\over 4}})\over 2^{1/4}\pi}$	(3)
$\displaystyle \varphi(e^{-\pi})$	$\textstyle =$	$\displaystyle {\pi^{1/4}\over\Gamma({\textstyle{3\over 4}})}$	(4)

Ramanujan's two-variable theta function is defined by

$\begin{displaymath} f(a,b)\equiv \sum_{n=-\infty}^\infty a^{n(n+1)/2}b^{n(n-1)/2} \end{displaymath}$

(5)

for $\vert ab\vert< 1$ . It is a generalization of the function $\varphi(x)$

$\begin{displaymath} f(x,x)=\varphi(x) \end{displaymath}$

(6)

and satisfies

$\begin{displaymath} f(-1,a)=0 \end{displaymath}$

(7)

$\begin{displaymath} f(a,b)=f(b,a)=(-a;ab)_\infty(-b;ab)_\infty(ab;ab)_\infty \end{displaymath}$

(8)

$\begin{displaymath} f(-q)\equiv f(-q,-q^2) = \sum_{k=0}^\infty (-1)^kq^{k(2k-1)/2} +\sum_{k=1}^\infty (-1)^kq^{k(2k+1)/2} = (q;q)_\infty, \end{displaymath}$

(9)

where $(q)_\infty$ are q-Series.