Multidimensional Padé approximation of binomial functions: Equalities

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 Pad\'e approximation of binomial functions}, and call $G$ the Pad\'e 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 Pad\'e approximations of binomial functions to be perfect (a specific sort of independence used in applications).