On the convergence acceleration of some continued fractions

TL;DR

This paper extends a convergence acceleration method for continued fractions with polynomial coefficients, improving tail approximations and applying it to a broader class including sums of two such fractions, with examples involving constants and functions.

Contribution

It generalizes an iterative tail approximation method to a wider class of continued fractions with polynomial numerators and denominators.

Findings

Enhanced convergence acceleration for a broader class of continued fractions.

Successful application to mathematical constants and special functions.

Improved asymptotic accuracy of tail approximations.

Abstract

A well known method for convergence acceleration of continued fraction $\K(a_n/b_n)$ is to use the modified approximants $S_n(\omega_n)$ in place of the classical approximants $S_n(0)$, where $\omega_n$ are close to tails $f^{(n)}$ of continued fraction. Recently, author proposed a method of iterative character producing tail approximations whose asymptotic expansion's accuracy is improving in each step. This method can be applied to continued fractions $\K(a_n/b_n)$, where $a_n$, $b_n$ are polynomials in $n$ ($\deg a_n=2$, $\deg b_n\leq 1$) for sufficiently large $n$. The purpose of this paper is to extend this idea for the class of continued fractions $\K(a_n/b_n + a_n'/b_n')$, where $a_n$, $a_n'$, $b_n$, $b_n'$ are polynomials in $n$ ($\deg a_n=\deg a_n', \deg b_n=\deg b_n'$). We give examples involving such continued fraction expansions of some mathematical constants, as well as elementary and special functions.