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Parametric Evaluations of the Rogers-Ramanujan Continued Fraction [PDF]
International Journal of Mathematics and Mathematical Sciences, 2011 In this paper with the help of the inverse function of the singular moduli we evaluate the Rogers-Ranmanujan continued fraction and its first derivative.
Nikos Bagis
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The Rogers–Ramanujan continued fraction and its level 13 analogue
Journal of Approximation Theory, 2015 The authors study properties of the so-called `level 13 analogue' of the Rogers-Ramanujan continued fraction \[ {\mathcal R}(q)={q^{1/5}\over 1+}{q\over 1+}{q^2\over 1+}{q^3\over 1+\cdots}=q^{1/5}\prod_{j=1}^{\infty}\,(1-q^j)^{\left({j\over 5}\right)}, \] where \(\left({j\over p}\right)\) is the Legendre symbol, given by \[ \begin{multlined} R(q)=q ...
Shaun Cooper, Dongxi Ye
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Singular values of the Rogers-Ramanujan continued fraction
Ramanujan Journal, 2006 In his first letter to G.H. Hardy, S. Ramanujan recorded several identities associated with the Rogers-Ramanujan continued fraction \[ \begin{aligned} R(q) &= \frac{q^{1/5} }{ 1+ \frac{ q}{1+ \frac{q^2}{ 1+ \cdots}}}, \end{aligned} \] one of which is \[ R(e^{-2\pi}) = \sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}. \] On page 210 of his Lost Notebook,
A. Gee, M. Honsbeek
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Modularity of certain products of the Rogers–Ramanujan continued fraction
Ramanujan JournalWe study the modularity of the functions of the form r(τ)ar(2τ)b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength ...
Russelle Guadalupe
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The Rogers–Ramanujan continued fraction [PDF]
Journal of Computational and Applied Mathematics, 1999 This is a survey of results for Ramanujan's continued fraction, \({q^{1/5} \over 1+}{q \over 1+}{q^2 \over 1+ \cdots}\), including evaluations at explicit values and modular equations satisfied by this function. There are also some results for generalizations.
Soon-Yi Kang, Heng Huat Chan
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ORTHOGONAL POLYNOMIALS ASSOCIATED WITH THE ROGERS-RAMANUJAN CONTINUED FRACTION [PDF]
Pacific Journal of Mathematics, 1983 We characterize the symmetric orthogonal polynomials {Pn(x)} such that {Pn(qnx)} is also orthogonal. This leads to orthogonal polynomials related to the denominator polynomials of the continued fractions of Rogers, Ramanujan, and Carlitz. We establish the orthogonality relation for these polynomials and show that the function ΣQ q n zn/(q; q)n that ...
W. Al-salam, M. Ismail
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On the Generalized Rogers–Ramanujan Continued Fraction [PDF]
The Ramanujan Journal, 2003 The generalized Rogers-Ramanujan continued fraction is defined for \(| q|< 1\) and any complex \(a\) by \[ R(a,q)= {1\over 1}{\;\atop +} {aq\over 1}{\;\atop +} {aq^2\over 1}{\;\atop +} {aq^3\over 1}{\;\atop +}\cdots. \] The authors prove an asymptotic formula stated by Ramanujan for \(R(a,e^{-x})\) as \(x\to 0+\).
Bruce C. Berndt, A. Yee
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Some Values for the Rogers-Ramanujan Continued Fraction
Canadian Journal of Mathematics, 1995 AbstractIn his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining ...
Bruce C. Bernd, H. Chan
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On the divergence of the Rogers-Ramanujan continued fraction on the unit circle [PDF]
Transactions of the American Mathematical Society, 2001 This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0,1) be denoted by [0, e 1 (t), e 2 (t),...] and let the i-th convergent of this continued ...
J. Laughlin, D. Bowman
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THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS
Bulletin of the Australian Mathematical Society, 2020 We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations.
Shane Chern, Dazhao Tang
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