caa a22 4500 463178165 CHVBK 20180406164831.0 cr unu---uuuuu 170326e20070601xx s 000 0 eng 10.1007/s11139-006-0243-3 doi (NATIONALLICENCE)springer-10.1007/s11139-006-0243-3 On the dynamics of certain recurrence relations [Elektronische Daten] [D. Borwein, J. Borwein, R. Crandall, R. Mayer] In recent analyses [3, 4] 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≠ a = be i φ with cos 2φ ≠ 1) or (a 2 = b 2∊ (−∞, 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 κ 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. Springer Science + Business Media, LLC, 2006 Complex continued fractions nationallicence Dynamical systems nationallicence Arithmetic-geometric mean nationallicence Matrix analysis nationallicence Stability theory nationallicence Borwein D. Department of Mathematics, University of Western Ontario, N6A 5B7, London Ontario, Canada aut Borwein J. Faculty of Computer Science Dalhousie University, B3H 1W5, Halifax, NS, Canada aut Crandall R. Center for Advanced Computation, Reed College, 97202, Portland, Oregon aut Mayer R. Department of Mathematics, Reed College, 97202, Portland, Oregon aut The Ramanujan Journal Kluwer Academic Publishers 13/1-3(2007-06-01), 63-101 1382-4090 13:1-3<63 2007 13 11139 https://doi.org/10.1007/s11139-006-0243-3 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-006-0243-3 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Borwein D. Department of Mathematics, University of Western Ontario, N6A 5B7, London Ontario, Canada aut NATIONALLICENCE 700 1- Borwein J. Faculty of Computer Science Dalhousie University, B3H 1W5, Halifax, NS, Canada aut NATIONALLICENCE 700 1- Crandall R. Center for Advanced Computation, Reed College, 97202, Portland, Oregon aut NATIONALLICENCE 700 1- Mayer R. Department of Mathematics, Reed College, 97202, Portland, Oregon aut NATIONALLICENCE 773 0- The Ramanujan Journal Kluwer Academic Publishers 13/1-3(2007-06-01), 63-101 1382-4090 13:1-3<63 2007 13 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer