Compatibility of Continued Fraction Convergents with Padé Approximants

Abstract

A Pade approximant (PA) to a function f is a rational function P m ∕ Q n matching the power expansion of f at least up to the (m + n)th power. On the contrary, the convergents of the Stieltjes, Jacobi, or Thiele continued fractions (CF) of f define all PA of f. However, the convergents of general CF are not necessarily PA. In this work, we present the rules stating when the convergents of CF are consistent with PA. The similar problem of compatible transformations of a variable and a function applied to PA was studied in [1].