Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(4): 1221-1233

Printed July 1, 2013

https://doi.org/10.4134/BKMS.2013.50.4.1221

Copyright © The Korean Mathematical Society.

On some modular equations and their applications II

Dae Hyun Paek and Jinhee Yi

Busan National University of Education, Korea Science Academy of KAIST

Abstract

We first derive some modular equations of degrees 3 and 9 and present their concise proofs based on algebraic computations. We then use these modular equations to establish explicit relations and formulas for the parameterizations for the theta functions $\varphi$ and $\psi$. In addition, we find specific values of the parameterizations to evaluate some numerical values of the cubic continued fraction.

Keywords: theta functions, modular equations, continued fractions

MSC numbers: Primary 11F27, 33C90; Secondary 11F20, 33C05, 33C75