Experimental Mathematics

On the Ramanujan AGM Fraction, II: The Complex-Parameter Case

J. Borwein and R. Crandall

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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.

Article information

Experiment. Math. Volume 13, Issue 3 (2004), 287-295.

First available in Project Euclid: 22 December 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 44-A20
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 11J70: Continued fractions and generalizations [See also 11A55, 11K50]

Continued fractions theta functions elliptic integrals hypergeometric functions special functions complex variables


Borwein, J.; Crandall, R. On the Ramanujan AGM Fraction, II: The Complex-Parameter Case. Experiment. Math. 13 (2004), no. 3, 287--295. http://projecteuclid.org/euclid.em/1103749837.

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