The two-point Padé approximation problem and its Hankel vector

Abstract

The two-point Padé approximation problem is to find a ratio of two coprime polynomials with some constraints on their degrees to approximate a function whose power series expansions at the origin and at infinity are given. In this paper, we introduce the Hankel vector for the two-point Padé approximation problem and establish the intrinsic connections between the two-point Padé approximation problem and a certain Padé approximation problem at infinity determined by the Hankel vector of the former. These connections provide us with a new way to study the structural characteristics of the two-point Padé table and to deduce the three-term recursive relations for the numerators and denominators of three adjacent entries in the two-point Padé table.