Rational approximants to symmetric formal Laurent series (Q688106)

Let \(F(z): = c_ 0 + \sum_{k=1}^ \infty c_ k (z^ k + z^{-k})\) be a symmetric Laurent series. The purpose of this paper is to construct Padé type approximants for \(F(z)\) of the form \[ \{k-1/k\}_ F^ S(z) = {P(z) \over Q(z)} + {P(z^{-1}) \over Q (z^{-1})} \] where \(Q(z): = \prod_{i=1}^ k (1-x_ iz)\) is given, \(0<| x_ i |<1\), \((x_ i = x_ j\) is possible) and \(P(z)\) is a polynomial of degree \(k- 1\) to be constructed. The authors show that Brezinski's method can be generalized to cover this situation. By means of the transformation \(z=t + \sqrt {t^ 2-1}\) they also find Padé-Chebyshev approximants to a Chebyshev series by the same method.