Closed form continued fraction expansions of special quadratic irrationals (Q1952715)

Summary: We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation \(G_{n+1} = mG_n + \ell G_{n-1}\) with \(G_0 = 0\), \(G_1 = 1\), \(m\) a positive integer, and \(\ell = \pm 1\) (note that \(m = \ell = 1\) gives the Fibonacci numbers). Let \(\varphi_{m, \ell} = \lim_{n \to \infty} G_n/G_{n-1}\). We find simple closed form continued fraction expansions for \(\varphi^k_{m, \ell}\) for any integer \(k\) by exploiting elementary properties of the recurrence relation and continued fractions.