On the Ramanujan AGM fraction. I: The real-parameter case. (English) Zbl 1090.11005

The Ramanujan AGM (Arithmetic Geometric Mean) continued fraction \[ \mathcal{R}_\eta(a,b):=\frac a\eta\,{\quad\atop +}\,\frac{(1b)^2}\eta\,{\quad\atop +}\,\frac{(2a)^2}\eta\,{\quad\atop +}\,\frac{(3b)^2}\eta\,{\quad\atop +}\,\frac{(4a)^2}\eta\,{\quad\atop +}\,\frac{(5b)^2}\eta\,{\quad\atop +\cdots} \] converges and has the surprising property that \[ \mathcal{R}_\eta\left(\frac{a+b}2,\,\sqrt{ab}\right)=\frac 12\{\mathcal{R}_\eta(a,b)+\mathcal{R}_\eta(b,a)\} \] in some domain \(D\subseteq\mathbb C^3\) containing the set where \(a>0\), \(b>0\) and \(\eta>0\). The convergence of \(\mathcal{R}_\eta(a,b)\) is however rather slow. The authors prove that \[ \left| \frac a{\mathcal{R}_1(a,b)}-f_n\right| <\begin{cases} 2nb^4(a/b)^n\,\,\,\text{for}\,\,0<a<b,\\ n(b/a)^{n+1}\,\,\,\text{for}\,\,0<b<a,\\ c\cdot n^h\,\,\,\text{for}\,\,0<b=a, \end{cases} \] where \(c\) and \(h\) are constants depending on \(a\). The purpose of this very nice paper is to find a faster way to compute \(\mathcal{R}_\eta(a,b)\) for real parameters \(a\), \(b\) and \(\eta\). The continued fraction diverges for \(\eta=0\). Hence an equivalence transformation shows that it suffices to consider the case where \(\eta=1\). The cases where \(a=0\) and/or \(b=0\) are trivial, so one may also assume that \(a\) and \(b\) are positive. The authors prove that then \(\mathcal{R}_1(a,b)\) can be computed by means of a series involving the standard elliptic integral \(K(k)\) with \(k:=\min\{\frac ab,\,\frac ba\}\) when \(a\neq b\). The bulk of the paper is devoted to the computation of \(\mathcal{R}(a):=\mathcal{R}_1(a,a)\) for \(a>0\). As a byproduct (!) they obtain a number of interesting expressions for \(\mathcal{R}(a)\), such as for instance \[ \mathcal{R}(a)=\frac{2a}{1+a}-\mathcal{R}\left(\frac a{1+2a}\right), \]
\[ \mathcal{R}(a)=2\sum_{k=0}^\infty \eta(k+1)(-1/a)^k\quad\text{where}\,\,\,\eta(s):=\frac 1{1^s}-\frac 1{3^2}+\frac 1{5^2}-\cdots, \] and a finite expression for \(\mathcal{R}(a)\) when \(a\) is rational which for instance leads to \[ \mathcal{R}(1)=\log 2,\quad\mathcal{R}(\tfrac 14)=\tfrac\pi 2-\tfrac 43,\quad \mathcal{R}(\tfrac 32)=\pi+\sqrt{3}\log(2-\sqrt{3}). \]


11A55 Continued fractions
40A05 Convergence and divergence of series and sequences
40A15 Convergence and divergence of continued fractions
Full Text: DOI Euclid


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