Closed Form Continued Fraction Expansions of Special Quadratic Irrationals. (17th January 2013) - Fishman, Daniel; Miller, Steven J.
Abstract : We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation G n + 1 = m G n + ℓ G n - 1 with G 0 = 0, G 1 = 1, m a positive integer, and ℓ = ± 1 (note that m = ℓ = 1 gives the Fibonacci numbers). Let φ m, ℓ = lim n → ∞ G n / G n - 1 . We find simple closed form continued fraction expansions for φ m, ℓ k for any integer k by exploiting elementary properties of the recurrence relation and continued fractions.