On a Khovanskii transformation for continued fractions (Q1822704)

Khovanskii introduced a transformation of continued fractions \(b_ 0+K^{\infty}_{i=1}(a_ i/1)\) with approximants \(c_ n=b_ 0+K^ n_{i=1}(a_ i/1).\) The present author shows that the approximants \(k_ n\) of Kohvanskij transformed continued fraction \(d_ 0+K^{\infty}_{i=1}(d_ i/1)\) satisfy \(K_ 0=C_ 0-1\), \(K_{2n}=(C_{2n}-F_ nC_{2n-1})/(1-F_ n),\quad K_{2n+1}=C_{2n+1}\) where \(F_ n=(C_{2n}-C_{2n-1})/(C_{2n-1}- C_{2n-2}).\) Interesting theorems, examples and remarks on this transformation are shown with application to convergence acceleration.