Convergence acceleration of limit periodic continued fractions under asymptotic side conditions (Q1093304)

Limit k-periodic continued fractions converging to a finite value can be accelerated by using modified approximants. As modifying factors are chosen the tail values of the corresponding periodic continued fraction. The method is simple and takes almost no extra operations. Under some easy to check side conditions this method is improved. The regular C- fraction expansions of hypergeometric functions \({}_ 2F_ 1(a,1;c;z)\) and \({}_ 2F_ 1(a,b;c;z)/_ 2F_ 1(a,b+1;c+1;z)\) are examples of continued fractions satisfying these conditions.