• The Pure and Applied Mathematics
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• v.28 no.4
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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.

• The Pure and Applied Mathematics
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• v.29 no.3
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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.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.97-107
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• 2007

Ramanujan gave six identities involving ${\theta}$-function expansion. We extended these identities by introducing a free parameter. We also give the interpretation of one of the identities using theory of partitions.

• The Pure and Applied Mathematics
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• v.28 no.1
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• pp.27-42
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• 2021

In this paper, we exploit some known theta function identities involving two parameters ��k,n and ��′k,n for the theta function �� to find about 54 new values of the Ramanujan's cubic continued fraction.

• East Asian mathematical journal
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• v.31 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.987-1028
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• 2014

In a recent systematic study, C. Sandon and F. Zanello offered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for multipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ramanujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.

• Kyungpook Mathematical Journal
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• v.49 no.3
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• pp.435-450
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• 2009

On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V ($q^3$) and also establish some explicit evaluations using the values of remarkable product of theta-function.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.937-956
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• 2018

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm [26] to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.1043-1059
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• 2019

In this paper, we use some theta function identities involving two parameters h_n,k and h'_n,k for the theta function φ to establish new evaluations of Ramanujan's cubic continued fraction.

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.767-777
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• 2007

By proving a summation formula, we enumerate the expansions for the mock theta functions of order 2 in terms of partial mock theta functions of order 2, 3 and 6. We show a relation between Ramanujan's ${\mu}(q)$-function and his sixth order mock theta functions. In addition, we also give the continued fraction representation for ${\mu}(q)$ and 2nd order mock theta functions and $Pad\acute{e}$ approximants.