@article {10.3844/jmssp.2005.142.145,
article_type = {journal},
title = {On the Relations among Characteristic Functions of Theta Functions},
author = {Yildiz, Ismet and Uyanik, Neslihan},
volume = {1},
year = {2005},
month = {Jun},
pages = {142-145},
doi = {10.3844/jmssp.2005.142.145},
url = {https://thescipub.com/abstract/jmssp.2005.142.145},
abstract = {In this study, using the characteristic values $\begin{bmatrix} \varepsilon\\ \varepsilon' \end{bmatrix} = \begin{bmatrix} 1\\ 1 \end{bmatrix}, \begin{bmatrix} 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1 \end{bmatrix}, \begin{bmatrix} 0\\ 0 \end{bmatrix} \pmod 2$ a theorem on the $\frac{1}{2^r}$ coefficients of periods of first order theta function according to the $(1,τ)$ period pair (for $r \in N^+$) is established. The following equalities are also obtained.

$\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{1}{2^r}-\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
$\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{\pi i}{2^r}}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
},
journal = {Journal of Mathematics and Statistics},
publisher = {Science Publications}
}