A method of convergence acceleration of some continued fractions II

Most of the methods for convergence acceleration of continued fractions K(a _m /b _m ) are based on the use of modified approximants S _m (ω _m ) in place of the classical ones S _m (0), where ω _m are close to the tails f ^(m) of the continued fraction. Recently (Nowak, Numer Algorithms 41(3):297–317, 2006), the author proposed an iterative method producing tail approximations whose asymptotic expansion accuracies are being improved in each step. This method can be successfully applied to a convergent continued fraction K(a _m /b _m ), where $a_m = alpha_{-2} m^2 + alpha_{-1} m + ldots$ , b _m ?=?β _???1 m?+?β ₀?+?... (α _???2?≠?0, $|beta_{-1}|^2+|beta_{0}|^2neq 0$ , i.e. $deg a_m=2$ , $deg b_min{0,1}$ ). The purpose of this paper is to extend this idea to the class of two-variant continued fractions K (a _n /b _n ?+?a _n ′/b _n ′) with a _n , a _n ′, b _n , b _n ′ being rational in n and $deg a_n=deg a_n'$ , $deg b_n=deg b_n'$ . We give examples involving continued fraction expansions of some elementary and special mathematical functions.