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.