On the convergence acceleration of some continued fractions

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 $n$ ($°a_n=2$, $°b_n\le 1$) 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 $a_n$, $a_n^{\prime }$, $b_n$, $b_n^{\prime }$ are polynomials in $n$ ($°a_n=°a_n^{\prime },°b_n=°b_n^{\prime }$). We give examples involving such continued fraction expansions of some mathematical constants, as well as elementary and special functions.

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