# On the dynamics of certain recurrence relations

Borwein, Jonathan M. and Crandall, Richard E. and Borwein, David and Mayer, Raymond (2007) On the dynamics of certain recurrence relations. Ramanujan Journal, 13 . pp. 63-101.

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In previous analyses \cite{borcra1, borcra2} the remarkable AGM continued fraction of Ramanujan---denoted ${\cal R}_1(a,b)$---was proven to converge for almost all complex parameter pairs $(a,b)$. It was conjectured that ${\cal R}_1$ diverges if and only if ($0 \not= a = b e^{i\phi}$ with $\cos^2 \phi \not= 1$) or ($a^2 = b^2 \in (-\infty,0)$). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for ${\cal R}_1$. This is accomplished by analyzing, using various special functions, the dynamics of sequences such as $(t_n)$ satisfying a recurrence $$t_n = (t_{n-1} + (n-1) \kappa_{n-1} t_{n-2})/n,$$ where $\kappa_n := a^2, b^2$ as $n$ be even, odd respectively. As a byproduct, we are able to give, in some cases, exact expressions for the $n$-th convergent to the fraction ${\cal R}_1$, thus establishing some precise convergence rates. It is of interest that this final resolution of convergence depends on rather intricate theorems for complex-matrix products, which theorems evidently being extensible to more general continued fractions.