Two-point Padé approximation and patterns in continued fractions for $(F_{n+1}/F_n)^k$

Abstract

We explore the simple continued fraction expansion for the $k$th power of the Fibonacci convergent $F_{n+1}/F_n$, as $n$ grows. For most $k$, this continued fraction consists of $k$ linearly growing \emph{patches} of infinite periodic continued fractions separated by \emph{junctions} that vary periodically in $n$. But under special conditions, two patches merge or, more strikingly, an exponentially growing \emph{monster term} replaces two patches and the junction between them. We trace these phenomena to properties of the Padé approximants that approximate $x^k$ near $x = ϕ$ and $x = \barϕ$ simultaneously. We prove the patterns for second-order unimodular recurrences and compute several more instances of this rare and apparently previously unnoticed monster term.