Truncation error bounds for limit-periodic continued fractions \(K(a_ n/1)\) with \(\lim a_ n=0\)
Truncation error bounds are developed for continued fractions \(K(a_ n/1)\) where \(| a_ n| \leq 1/4\) for all n sufficiently large. The bounds are particularly suited (some are shown to be best) for the limit- periodic case when lim \(a_ n=0\). Among the principal results is the following: If \(| a_ n| \leq \alpha /n^ p\) for all n sufficiently large (with constants \(\alpha >0\), \(p>0)\), then \(| f- f_ m| \leq C[D/(m+2)]^{p(m+2)}\) for all m sufficiently large (for some constants \(C>0\), \(D>0)\). Here f denotes the limit (assumed finite) of \(K(a_ n/1)\) and \(f_ m\) denotes its mth approximant. Applications are given for continued fraction expansions of ratios of Kummer functions \(_ 1F_ 1\) and of ratios of hypergeometric functions \({}_ 0F_ 1\). It is shown that \(p=1\) for \(_ 1F_ 1\) and \(p=2\) for \(_ 0F_ 1\), where p is the parameter determining the rate of convergence. Numerical examples indicate that the error bounds are indeed sharp.
scientific article; zbMATH DE number 3899472
scientific article