Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.13/928084
- On the dynamics of certain recurrence relations
- In recent analyses 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 iφ 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₋₋₁ᵗtn₋₂)/n, where κn := a², b² 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 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.
- The Ramanujan Journal Vol. 13, Issue 1-3, p. 63-101
- Publisher Link
complex continued fractions;
- Resource Type
- journal article