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  • P-ISSN3059-0604
  • E-ISSN3059-1309
  • KCI

On evaluations of the cubic continued fraction by modular equations of degree 9

Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics / Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics, (P)3059-0604; (E)3059-1309
2016, v.23 no.3, pp.223-236
https://doi.org/10.7468/jksmeb.2016.23.3.223
PAEK, DAE HYUN
YI, JINHEE

Abstract

We show how to evaluate the cubic continued fraction <TEX>$G(e^{-{\pi}\sqrt{n}})$</TEX> and <TEX>$G(-e^{-{\pi}\sqrt{n}})$</TEX> for n = 4<sup>m</sup>, 4<sup>&#x2212;m</sup>, 2 &#xB7; 4<sup>m</sup>, and 2<sup>&#x2212;1</sup> &#xB7; 4<sup>&#x2212;m</sup> for some nonnegative integer m by using modular equations of degree 9. We then find some explicit values of them.

keywords
continued fraction, modular equations, theta functions

Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics