On the dynamics of certain recurrence relations

In recent analyses [3, 4] the remarkable AGM continued fraction of Ramanujan—denoted R1(a, b)—was proven to converge for almost all complex parameter pairs (a, b). It was conjectured that R1 diverges if and only if (0 = a = be with cos2 φ = 1) or (a2 = b2 ∈ (−∞, 0)). In the present treatment we resolve this conjecture to the positive, thus establishing the precise convergence domain for R1. This is accomplished by analyzing, using various special functions, the dynamics of sequences such as (tn) satisfying a recurrence tn = (tn−1 + (n − 1)κn−1tn−2)/n, where κn := a2, b2 as n be even, odd respectively. Research supported by NSERC. Research supported by NSERC, the Canada Foundation for Innovation and the Canada Research Chair Program. D. Borwein ( ) Department of Mathematics, University of Western Ontario, London Ontario Canada N6A 5B7 e-mail: dborwein@uwo. ca J. Borwein Faculty of Computer Science Dalhousie University, Halifax NS Canada B3H 1W5 e-mail: [email protected] R. Crandall Center for Advanced Computation, Reed College, Portland, Oregon 97202 e-mail: [email protected] R. Mayer Department of Mathematics, Reed College, Portland Oregon 97202 e-mail: [email protected] Springer 64 D. Borwein et al. As a byproduct, we are able to give, in some cases, exact expressions for the n-th convergent to the fraction R1, 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.