Multidimensional Padé approximation of binomial functions: Equalities

Abstract

Let ${\omega }_0,\dots ,{\omega }_M$ be complex numbers. If $H_0,\dots ,H_M$ are polynomials of degree at most ${\rho }_0,\dots ,{\rho }_M$, and $G(z)={\sum }_{m=0}^M{H}_m(z)(1-z{)}^{{\omega }_m}$ has a zero at $z=0$ of maximal order (for the given ${\omega }_m,{\rho }_m$), we say that $H_0,\dots ,H_M$ are a \emph{multidimensional Padé approximation of binomial functions}, and call $G$ the Padé remainder. We collect here with proof all of the known expressions for $G$ and $H_m$, including a new one: the Taylor series of $G$. We also give a new criterion for systems of Padé approximations of binomial functions to be perfect (a specific sort of independence used in applications).

30 pages, ancillary Mathematica notebook