Research Article Open Access

On the Relations among Characteristic Functions of Theta Functions

Ismet Yildiz and Neslihan Uyanik

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.

  1. $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{1}{2^r}-\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 1 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$
  2. $\exp\left\{{-\frac{1}{{{4^r}}}\left({\tau+2}\right)\pi i-\frac{\pi i}{2^r}}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 1 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)=\exp\left\{{-\frac{1}{4^r}(\tau+2)\pi i}\right\}.\theta\left[\begin{array}{l} 0 + \frac{1}{2^{r-1}}\\ 0 + \frac{1}{2^{r-1}}\\ \end{array}\right](0,\tau)$

Journal of Mathematics and Statistics
Volume 1 No. 2, 2005, 142-145

DOI: https://doi.org/10.3844/jmssp.2005.142.145

Published On: 30 June 2005

How to Cite: Yildiz, I. & Uyanik, N. (2005). On the Relations among Characteristic Functions of Theta Functions. Journal of Mathematics and Statistics, 1(2), 142-145. https://doi.org/10.3844/jmssp.2005.142.145

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Keywords

  • First Order Theta Function
  • Characteristic Values