The Rogers-Ramanujan continued fraction and a quintic iteration for $1/\pi$

Chan, Heng Huat; Cooper, S. Barry; Liaw, Wen-Chin

doi:10.1090/s0002-9939-07-09031-4

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…Although it is too long to be written down, the closed form has the benefit that it avoids the loss in accuracy associated with iterated numerical computation. Other methods for calculating the digits of π include Chan et al's [11] quintic approximation for 1/π; the series for 1/π obtained by the Borweins [9], which yields 50 digits of π per term; and Berndt and Chan's series [6], which obtains about 73 or 74 digits of π per term. If we forgo closed-form expressions, we can use modular relations to obtain greater approximation accuracy in each step of a numerical iteration.…”

Section: R(q) and Approximations To πmentioning

confidence: 99%

Properties of reciprocity formulas for the Rogers–Ramanujan continued fractions

Kohli

2019

Ramanujan J

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Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fractions. Two reciprocity formulas each are also associated with the Ramanujan-Göllnitz-Gordon continued fractions and a level-13 analog of the Roger-Ramanujan continued fractions. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients of the quadratic equations to important formulas obtained by Ramanujan, examine a pattern in the relation between a continued fraction and its parameters, and use the reciprocity formulas to obtain close approximations for all values of the continued fractions. We highlight patterns in the expressions for certain explicit values of the Rogers-Ramanujan continued fractions by expressing them in terms of the golden ratio. We extend the analysis to reciprocity formulas for Ramanujan's cubic continued fraction and the Ramanujan-Selberg continued fraction.Roger-Ramanujan continued fractions and Ramanujan-Göllnitz-Gordon continued fractions and Ramanujan's cubic continued fractions and Ramanujan-Selberg continued fraction and Reciprocal formulas Date

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Section: R(q) and Approximations To πmentioning

confidence: 99%

Properties of reciprocity formulas for the Rogers–Ramanujan continued fractions

Kohli

2019

Ramanujan J

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“…Iterations. In this section we use the ideas from [14], that were utilized in [11], to produce four iterations that converge to 1/π. One of these (the case n = 2) is new.…”

mentioning

confidence: 99%

“…The quartic iteration for n = 4 turns out to be equivalent to one given by Chan in [10, Iteration 1.6]. The quintic iteration for n = 5 was given by Chan, Cooper and Liaw in [11]. (1 − q 8j−7 )(1 − q 8j−1 ) (1 − q 8j−5 )(1 − q 8j−3 ) and define v = v(q) = u(q 2 ), w = w(q) = u 2 (q).…”

mentioning

confidence: 99%