Accelerations of generalized Fibonacci sequences

TL;DR

This paper develops methods to accelerate convergence of ratios in generalized Fibonacci sequences, using recurrent formulas to generate subsequences and applying classical approximation methods, with applications to continued fractions for quadratic irrationals.

Contribution

It introduces recurrent formulas for subsequences of ratios in generalized Fibonacci sequences and links these to classical approximation methods, enhancing convergence analysis.

Findings

Recurrent formulas for subsequences of ratios are established.

Classical methods generate subsequences with accelerated convergence.

Applications to continued fractions for quadratic irrationalities are demonstrated.

Abstract

In this paper we study how to accelerate the convergence of the ratios (x_n) of generalized Fibonacci sequences. In particular, we provide recurrent formulas in order to generate subsequences (x_{g_n}) for every linear recurrent sequence (g_n) of order 2. Using these formulas we prove that some approximation methods, as secant, Newton, Halley and Householder methods, can generate subsequences of (x_n). Moreover, interesting properties on Fibonacci numbers arise as an application. Finally, we apply all the results to the convergents of a particular continued fraction which represents quadratic irrationalities.