• 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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제29권3호
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• pp.245-254
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• 2022

In this paper, we use some theta-function identities involving certain parameters to show how to evaluate Rogers-Ramanujan continued fraction R($e^{-2{\pi}\sqrt{n}}$) and S($e^{-{\pi}\sqrt{n}}$) for $n=\frac{1}{5.4^m}$ and $\frac{1}{4^m}$, where m is any positive integer. We give some explicit evaluations of them.

• East Asian mathematical journal
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• 제31권5호
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• pp.659-665
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• 2015

Combining and specializing some known results, we establish six identities which depict six modular relations for the Roger-Ramanujan type identities and two equivalent representations for Jacobian identity expressed in terms of combinatorial partition identities and Ramanujan-Selberg continued fraction. Two q-product identities are also considered.

• 대한수학회보
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• 제55권2호
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• pp.449-467
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• 2018

In this paper we introduce and study the lattice paths for which the horizontal step is allowed at height $h{\geq}0$, $h{\in}{\mathbb{Z}}$. By doing so these paths generalize the heavily studied weighted lattice paths that consist of horizontal steps allowed at height zero only. Six q-series identities of Rogers-Ramanujan type are studied combinatorially using these generalized lattice paths. The results are further extended by using (n + t)-color overpartitions. Finally, we will establish that there are certain equinumerous families of (n + t)-color overpartitions and the generalized lattice paths.

• 대한수학회지
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• 제44권1호
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• pp.55-107
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• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}$. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots$ ([1]) is transcendental.

• 대한수학회논문집
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• 제32권4호
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• pp.885-898
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• 2017

The Bailey lemma has been a powerful tool in the discovery of identities of Rogers-Ramanujan type and also ordinary and basic hyper-geometric series identities. The mechanism of Bailey lemma has also led to the concepts of Bailey pair and Bailey chain. In the present work certain new WP-Bailey pairs have been established. We also have deduced a number of basic hypergeometric series identities as an application of new WP-Bailey pairs.

• 대한수학회보
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• 제59권1호
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• pp.15-25
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• 2022

In 1963, Graham introduced a problem to find integer partitions such that the reciprocal sum of their parts is 1. Inspired by Graham's work and classical partition identities, we show that there is an integer partition of a sufficiently large integer n such that the reciprocal sum of the parts is 1, while the parts satisfy certain congruence conditions.

• 대한수학회지
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• 제45권5호
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• pp.1379-1391
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• 2008

Let k be an imaginary quadratic field, ${\eta}$ the complex upper half plane, and let ${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$. For n, t ${\in}{\mathbb{Z}}^+$ with $1{\leq}t{\leq}n-1$, set n=${\delta}{\cdot}2^{\iota}$(${\delta}$=2, 3, 5, 7, 9, 13, 15) with ${\iota}{\geq}0$ integer. Then we show that $q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$ are algebraic numbers.