The multipoint Padé table and general recurrences for rational interpolation

We first review briefly the Newton-Padé approximation problem and the analogous problem with additional interpolation conditions at infinity, which we call multipoint Padé approximation problem. General recurrence formulas for the Newton-Padé table combine either two pairs of Newton-Padé forms or one such pair and a pair of multipoint Padé forms. We show that, likewise, certain general recurrences for the multipoint Padé table compose two pairs of multipoint Padé forms to get a new pair of multipoint Padé forms. We also discuss the possibility of superfast, i.e.,O(n log² n) algorithms for certain rational interpolation problems.