Computation of the \(c\)-table related to the Padé approximation (Q1789779)

Summary: The aim of this paper is to give a complete and practical method for numerical application of Padé approximation with the help of the \(c\)-table analysis. We present an exhaustive list of useful formulas to compute a \(c\)-table related to a formal power series \(C(z)=\sum^\infty_{n=0}c_nz^n\). Some of these formulas are not widely known, because they were presented in publications of limited circulation. Some others were never published, as three symmetric Paszkowski-like formulas to overcome the blocks in a \(c\)-table or an extension of local error formula for Padé approximants in the blocks. All formulas are given in two versions: in terms of Toeplitz determinants (\(c\)-table) and in the version of Hankel determinants (\(c\)-table). We compare the theory with numerical observations by reproducing different computational aspects of software producing the \(c\)-tables with the presence of blocks and their evolution following the evolution of computer environment.