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Two-point Padé approximation and patterns in continued fractions for (Fn+1/Fn)k
Abstract
We explore the simple continued fraction expansion for the kth power of the Fibonacci convergent Fn+1/Fn, 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 xk near x = ϕ and x = ϕ̅ simultaneously. We prove the patterns for second-order unimodular recurrences and compute several more instances of this rare and apparently previously unnoticed monster term.
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BibTeX
@misc{odorney2026twopoint,
title = {{Two-point Padé approximation and patterns in continued fractions for $(F_n+1/F_n)^k$}},
author = {Evan M. O'Dorney},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01029},
eprint = {2611.01029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01029},
}
RIS
TY - UNPB
TI - Two-point Padé approximation and patterns in continued fractions for (F_n+1/F_n)^k
AU - Evan M. O'Dorney
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01029
UR - https://arxiv.org/abs/2611.01029
AB - We explore the simple continued fraction expansion for the kth 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 = ϕ̅ simultaneously. We prove the patterns for second-order unimodular recurrences and compute several more instances of this rare and apparently previously unnoticed monster term.
ER -