A study on ratio of the H–Theta functions M. Nuri K€ ult € ur, Abdullah Kaplan * Department of Mathematics, Kazım Karabekir Education Faculty, € U niversity of Atat€ urk, Erzurum 25240, Turkey Abstract In this work, a non-linear differential equation which contains the squares of the theta functions and also satisfied by wðzÞ¼ h 1 ðzÞ=h 3 ðzÞ is established by using known properties of the theta functions h 1 , h 2 , h 3 and h 4 . Ó 2003 Elsevier Inc. All rights reserved. Keywords: Theta functions; Elliptical functions 1. Introduction Definition 1.1. A lattice of X of complex numbers is an aggregate of complex numbers with the two properties: i(i) X is a group with respect to addition, (ii) The absolute magnitudes of the non-zero elements are bounded below, i.e., there is a real number k > 0 such that jwj P k for all w 6¼ 0 in X [2]. The set X ¼fmw 1 þ nw 2 : m; n 2 Z g is a 2-dimensional lattice, where w 1 and w 2 are linearly independent complex numbers. The pair ðw 1 ; w 2 Þ which generates the lattice is called a basis of the lattice. The periodic function f ðzÞ is called doubly-periodic if its period lattice has dimension 2. * Corresponding author. E-mail address: akaplan@atauni.edu.tr (A. Kaplan). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.091 Applied Mathematics and Computation 158 (2004) 353–358 www.elsevier.com/locate/amc