• Kyungpook Mathematical Journal
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• 제51권4호
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• pp.419-427
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• 2011

In this paper, making use of transformation due to S. N. Singh [21], an at-tempt has been made to establish certain results involving basic bilateral hypergeometric series and continued fractions.

• 한국수학사학회지
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• 제13권2호
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• pp.49-64
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• 2000

Every real number can be expressed as a simple continued fraction. In particular, a number is rational if and only if its simple continued fraction has a finite number of terms. Owing to this property, continued fractions have been a powerful tool which determines a real number to be rational or not. Continued fractions provide not only a series of best estimate for a real number, but also a useful method for finding near commensurabilities between events with different periods. In this paper, we investigate the history and some properties of continued fractions, and then consider their applications in several examples. Also we explain why the Fibonacci numbers and the Golden section appear in nature in terms of continued fractions, with some examples such as the arrangements of petals round a flower, leaves round branches and seeds on seed head.

• 대한수학회지
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• 제48권5호
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• pp.887-898
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• 2011

In this paper, we first discuss the convergence of the continued fractions of generalized Rogers-Ramanujan type in the modified sense. Then we prove several equalities concerning these continued fractions. The proofs of our main results are mainly based on the Bauer-Muir transformation.

• 대한수학회지
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• 제60권2호
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• pp.395-406
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• 2023

As an analogy of the Rogers-Ramanujan continued fraction, we define a Ramanujan continued fraction of order eighteen. There are essentially three Ramanujan continued fractions of order eighteen, and we study them using the theory of modular functions. First, we prove that they are modular functions and find the relations with the Ramanujan cubic continued fraction C(𝜏). We can then obtain that their values are algebraic numbers. Finally, we evaluate them at some imaginary quadratic quantities.

• Korean Journal of Mathematics
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• 제25권2호
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• pp.137-145
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• 2017

We consider some simple periodic functions on the field of rational numbers with values in ${\mathbb{Q}}/{\mathbb{Z}}$ which are defined in terms of lowest-term-expression of rational numbers. We prove the density of graphs of these functions by constructing explicitly points on the graphs close to a given point using continued fractions.

• 충청수학회지
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• 제27권2호
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• pp.183-203
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• 2014

We examine fundamental units of quadratic function fields from continued fraction of $\sqrt{D}$. As a consequence, we give another proof of geometric analog of Ankeny-Artin-Chowla-Mordell conjecture and bounds for class number, and study real quadratic function fields of minimal type with quasi-period 4.

• 대한수학회논문집
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• 제22권3호
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• pp.331-351
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• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k$, $q=e^{{\pi}i\tau}$. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.173-184
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• 2016

Mock theta functions are the most interesting topic mentioned in Ramanujan's Lost Notebook, due to its emerging application in the field of Number theory, Quantum invariants theory and etc. In the present research articles we have made an attempt to develop continued fractions representation of all the existing Mock theta functions.

• 대한수학회지
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• 제56권4호
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• pp.1049-1061
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• 2019

For ${\alpha}{\geq}1$, let $s_{\alpha}(n)={\lceil}{\alpha}n{\rceil}-{\lceil}{\alpha}(n-1){\rceil}$. A continued fraction $C({\alpha})=[0;s_{\alpha}(1),s_{\alpha}(2),{\ldots}]$ is considered and analyzed. Appealing to Diophantine approximation, we investigate the differentiability of $C({\alpha})$, and then show its singularity.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제28권4호
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• pp.377-386
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• 2021

In this paper, we use theta-function identities involving parameters 𝑙_5,n, 𝑙'_5,n, and 𝑙'_5,4n to evaluate the Rogers-Ramanujan continued fractions $R(e^{-2{\pi}{\sqrt{n/20}}})$ and $S(e^{-{\pi}{\sqrt{n/5}}})$ for some positive rational numbers n.