On the dynamics of certain recurrence relations

Abstract

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.