• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.273-280
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• 2005

We generalize a continued fraction of Ramanujan by introducing a free parameter. We give the closed form for the continued fraction. We also consider the finite form giving $n^{th}$ convergent using partition theory.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.367-373
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• 2016

Adiga and Anitha [1] investigated the Ramanujan's continued fraction (18) to present many interesting identities. Motivated by this work, by using known formulas, we also investigate the Ramanujan's continued fraction (18) to give certain relationships between the Ramanujan's continued fraction and the combinatorial partition identities given by Andrews et al. [3].

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.155-159
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• 2009

Using the Binet's formula, we show that the quotient related ratio $l_{1(x)}\;\neq\;0$ for the eventually periodic continued fraction x. Using this ratio, we also show that the derivative of the Minkowski question mark function at the simple periodic continued fraction is infinite or 0. In particular, $l_1({[\bar{1}]})$ = 2 log $\gamma$ where $\gamma$ is the golden mean $(1+\sqrt{5})/2$ and the derivative of the Minkowski question mark function at the simple periodic continued fraction $[\bar{1}]$ is infinite.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.319-332
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• 2018

In the present paper, we establish relationship between continued fraction U(-q) of order 12 and Ramanujan's cubic continued fraction G(-q) and $G(q^n)$ for n = 1, 2, 3, 5 and 7. Also we evaluate U(q) and U(-q) by using two parameters for Ramanujan's theta-functions and their explicit values.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.29 no.3
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• pp.387-399
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• 2014

The ratios of any two Fibonacci numbers are expressed by means of semisimple continued fraction.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.603-607
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• 2005

We give an integral representation for an analogous continued fraction of Ramanujan, for this we first prove an interesting identity.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.699-709
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• 2016

Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1623-1630
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• 2013

The aim of this paper is to prove that every quadratic formal power series ${\omega}$ can be expressed as a periodic non-simple continued fraction having period length one.

• Journal of the Chungcheong Mathematical Society
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• v.18 no.1
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• pp.95-100
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• 2005

In this paper, we consider the finite form of the continued fraction found in the unorganized material, Entry 9 [2], and give the sum of n terms. We give the $n^{th}$ convergent series in two ways-one by simple expansion and the other by using partition theory.