Linear approximations to continued fractions \(K(z_ n/1)\)
The author considers continued fractions \(K^{\infty}_ 1(\frac{a_ n+\epsilon_ nz}{1})=F(z)\) where \(K^{\infty}_ 1(\frac{a_ n}{1})\) and its tails \(K^{\infty}_ N(\frac{a_ n}{1})\) are supposed known and converging to finite values. Under proper conditions on the variable quantities involved - the \(a_ n\) lie in a parabolic region - it is proven that F(z) is holomorphic in some disk \(| z| <1/\theta\) and F'(0) is evaluated as an infinite series which \(F(1)-K^{\infty}_ 1(\frac{a_ n}{1})\) is estimated. The proof is based on previous papers of the author. Applications to convergence acceleration and some examples are considered.
scientific article; zbMATH DE number 4081892