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Three Ramanujan continued fractions with modularity
- Three Ramanujan continued fractions with modularity
- Lee Y.; Park Y.K.
- Ewha Authors
- 이윤진; 박윤경
- SCOPUS Author ID
- 이윤진; 박윤경
- Issue Date
- Journal Title
- Journal of Number Theory
- Journal of Number Theory vol. 188, pp. 299 - 323
- Class field theory; Modular function; Ramanujan continued fraction
- Academic Press Inc.
- SCI; SCIE; SCOPUS
- Document Type
- We study three Ramanujan continued fractions c(τ),W(τ) and T(τ). In fact, c(τ) and W(τ) are modular functions of level 16, and T(τ) is a modular function of level 32. We first prove that the values of c(τ) and W(τ) can generate the ray class field modulo 4 over an imaginary quadratic field K. We also prove that 2/(1−c(τ)),1/W(τ),T(τ)+1/T(τ) are algebraic integers for any imaginary quadratic quantity τ. Furthermore, we find the modular equations of c(τ),T(τ) and W(τ) for any level, and we show that c(τ) and W(τ) satisfy the Kronecker's congruence. We can express the value c(rτ) (respectively, T(rτ),W(rτ)) in terms of radicals for any positive rational number r when the value c(τ) (respectively, T(τ),W(τ)) can be written as radicals. © 2018 Elsevier Inc.
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