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2004 On the Ramanujan AGM Fraction, II: The Complex-Parameter Case
J. Borwein, R. Crandall
Experiment. Math. 13(3): 287-295 (2004).


The Ramanujan continued fraction

{\small $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle \eta+{}_{\ddots}}}}}$$}

is interesting in many ways; e.g., for certain complex parameters $(\eta, a,b)$ one has an attractive AGM relation ${\cal R}_{\eta} (a,b) + {\cal R}_{\eta}(b,a) = 2 {\cal R}_{\eta} \left((a+b)/2, \sqrt{ab} \right)$. Alas, for some parameters the continued fraction ${\cal R}_{\eta}$ does not converge; moreover, there are converging instances where the AGM relation itself does not hold. To unravel these dilemmas we herein establish convergence theorems, the central result being that ${\cal R}_1$ converges whenever $|a| \not= |b|$. Such analysis leads naturally to the conjecture that divergence occurs whenever $a=b e^{i\phi}$ with $\cos^2\phi \not = 1$ (which conjecture has been proven in a separate work) [Borwein et al. ] We further conjecture that for $a/b$ lying in a certain---and rather picturesque---complex domain, we have both convergence and the truth of the AGM relation.


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J. Borwein. R. Crandall. "On the Ramanujan AGM Fraction, II: The Complex-Parameter Case." Experiment. Math. 13 (3) 287 - 295, 2004.


Published: 2004
First available in Project Euclid: 22 December 2004

zbMATH: 1122.11044
MathSciNet: MR2103327

Primary: 44-A20
Secondary: 11J70 , 33C05

Keywords: complex variables , continued fractions , elliptic integrals , hypergeometric functions , Special functions , Theta functions

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 3 • 2004
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