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Ramanujan Theta Functions

Ramanujan's one-variable theta function is defined by

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

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 $\varphi(x)$
\end{displaymath} (6)

and satisfies
\end{displaymath} (7)

\end{displaymath} (8)

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.

See also Jacobi Triple Product, q-Series, Schröter's Formula

© 1996-9 Eric W. Weisstein