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Three Ramanujan continued fractions with modularity

Title
Three Ramanujan continued fractions with modularity
Authors
Lee Y.Park Y.K.
Ewha Authors
이윤진박윤경
SCOPUS Author ID
이윤진scopus; 박윤경scopus
Issue Date
2018
Journal Title
Journal of Number Theory
ISSN
0022-314XJCR Link
Citation
Journal of Number Theory vol. 188, pp. 299 - 323
Keywords
Class field theoryModular functionRamanujan continued fraction
Publisher
Academic Press Inc.
Indexed
SCI; SCIE; SCOPUS WOS scopus
Document Type
Article
Abstract
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.
DOI
10.1016/j.jnt.2018.01.012
Appears in Collections:
자연과학대학 > 수학전공 > Journal papers
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